brown’s spectral measure, and the free multiplicative...
TRANSCRIPT
![Page 1: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/1.jpg)
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Brown’s Spectral Measure, and the Free Multiplicative Brownian Motion
West Coast Operator Algebras Seminar
Seattle University
Todd Kemp
UC San Diego
October 7, 2018
![Page 2: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/2.jpg)
Dedication
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
2 / 35
This talk, and all my work, is dedicated to the memory of my father:
Robin Edward Kemp September 9, 1938 – August 4, 2018
who was a brilliant, hard-working, gentle, and humble man, and is
the source of my strength, my intellect, and my success.
![Page 3: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/3.jpg)
Giving Credit where Credit is Due
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• Biane, P.: Free Brownian motion, free stochastic calculus and random
matrices. Fields Inst. Commun. vol. 12, Amer. Math. Soc., Providence, RI,
1-19 (1997)
• Biane, P.: Segal-Bargmann transform, functional calculus on matrix spaces
and the theory of semi-circular and circular systems. J. Funct. Anal. 144, 1,
232-286 (1997)
• Driver; Hall; K: The large-N limit of the Segal–Bargmann transform on UN .
J. Funct. Anal. 265, 2585-2644 (2013)
• K: The Large-N Limits of Brownian Motions on GLN . Int. Math. Res. Not.
IMRN, no. 13, 4012-4057 (2016)
• Collins, Dahlqvist, K: The Spectral Edge of Unitary Brownian Motion.
Probab. Theory Related Fields 170, no. 102, 49-93 (2018)
• Hall, K: Brown Measure and the Free Multiplicative Brownian Motion.
arXiv:1810.00153
![Page 4: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/4.jpg)
Brown’s Spectral Measure
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
4 / 35
![Page 5: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/5.jpg)
Brown’s Spectral Measure in Tracial von Neumann Algebras
5 / 35
If (A, τ) is a W ∗-probability space, then any normal operator a ∈ A has a
spectral measure µa = τ Ea. If A is a normal matrix, µA is its ESD. It is
characterized (nicely) by the ∗-distribution of a:
∫
C
zkzℓ µa(dzdz) = τ(aka∗ℓ).
![Page 6: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/6.jpg)
Brown’s Spectral Measure in Tracial von Neumann Algebras
5 / 35
If (A, τ) is a W ∗-probability space, then any normal operator a ∈ A has a
spectral measure µa = τ Ea. If A is a normal matrix, µA is its ESD. It is
characterized (nicely) by the ∗-distribution of a:
∫
C
zkzℓ µa(dzdz) = τ(aka∗ℓ).
If a is not normal, there is no such measure. But there is a substitute: Brown’s
spectral measure. Let L(a) denote the (log) Kadison–Fuglede determinant:
L(a) =
∫
R
log t µ|a|(dt) = τ
(∫
R
log t E|a|(dt)
)
![Page 7: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/7.jpg)
Brown’s Spectral Measure in Tracial von Neumann Algebras
5 / 35
If (A, τ) is a W ∗-probability space, then any normal operator a ∈ A has a
spectral measure µa = τ Ea. If A is a normal matrix, µA is its ESD. It is
characterized (nicely) by the ∗-distribution of a:
∫
C
zkzℓ µa(dzdz) = τ(aka∗ℓ).
If a is not normal, there is no such measure. But there is a substitute: Brown’s
spectral measure. Let L(a) denote the (log) Kadison–Fuglede determinant:
L(a) =
∫
R
log t µ|a|(dt) = τ
(∫
R
log t E|a|(dt)
)
= τ(log |a|)
(the last = holds if a−1 ∈ A).
![Page 8: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/8.jpg)
Brown’s Spectral Measure in Tracial von Neumann Algebras
5 / 35
If (A, τ) is a W ∗-probability space, then any normal operator a ∈ A has a
spectral measure µa = τ Ea. If A is a normal matrix, µA is its ESD. It is
characterized (nicely) by the ∗-distribution of a:
∫
C
zkzℓ µa(dzdz) = τ(aka∗ℓ).
If a is not normal, there is no such measure. But there is a substitute: Brown’s
spectral measure. Let L(a) denote the (log) Kadison–Fuglede determinant:
L(a) =
∫
R
log t µ|a|(dt) = τ
(∫
R
log t E|a|(dt)
)
= τ(log |a|)
(the last = holds if a−1 ∈ A). Then λ 7→ L(a− λ) is subharmonic on C, and
µa =1
2π∇2
λL(a− λ)
is a probability measure on C. If A is any matrix, µA is its ESD.
![Page 9: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/9.jpg)
The Circular Law
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
6 / 35
Consider a circular operator z (mentioned yesterday in Brent
Nelson’s talk):
z =1√2(x+ iy) x, y freely independent semicirculars.
It is not too difficult to compute from the definition that
µz = uniform probability measure on D.
![Page 10: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/10.jpg)
The Circular Law
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
6 / 35
Consider a circular operator z (mentioned yesterday in Brent
Nelson’s talk):
z =1√2(x+ iy) x, y freely independent semicirculars.
It is not too difficult to compute from the definition that
µz = uniform probability measure on D.
This goes hand in hand with the fact that z is the large-N limit (in
∗-distribution) of the Ginibre ensemble (all i.i.d. Gaussian entries),
whose ESD converges to the uniform probability measure on D
(that’s the Circular Law proved by Ginibre, Girko, Bai, Tao-Vu, . . . )
![Page 11: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/11.jpg)
The Circular Law
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
6 / 35
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
![Page 12: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/12.jpg)
The Circular Law
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
6 / 35
Consider a circular operator z (mentioned yesterday in Brent
Nelson’s talk):
z =1√2(x+ iy) x, y freely independent semicirculars.
It is not too difficult to compute from the definition that
µz = uniform probability measure on D.
This goes hand in hand with the fact that z is the large-N limit (in
∗-distribution) of the Ginibre ensemble (all i.i.d. Gaussian entries),
whose ESD converges to the uniform probability measure on D
(that’s the Circular Law proved by Ginibre, Girko, Bai, Tao-Vu, . . . )
However, the connection between limit ESD and Brown measure is
actually very complicated.
![Page 13: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/13.jpg)
Properties of Brown Measure
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
7 / 35
The Brown measure has some nice properties analogous to the
spectral measure, but not all:
• τ(ak) =
∫
C
zk µa(dzdz) and τ(a∗k) =
∫
C
zk µa(dzdz)
but you cannot max and match.
![Page 14: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/14.jpg)
Properties of Brown Measure
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
7 / 35
The Brown measure has some nice properties analogous to the
spectral measure, but not all:
• τ(ak) =
∫
C
zk µa(dzdz) and τ(a∗k) =
∫
C
zk µa(dzdz)
but you cannot max and match.
• τ(log |a− λ|) = L(a− λ) =
∫
C
log |z − λ|µa(dzdz) for
large λ, and this characterizes µa.
![Page 15: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/15.jpg)
Properties of Brown Measure
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
7 / 35
The Brown measure has some nice properties analogous to the
spectral measure, but not all:
• τ(ak) =
∫
C
zk µa(dzdz) and τ(a∗k) =
∫
C
zk µa(dzdz)
but you cannot max and match.
• τ(log |a− λ|) = L(a− λ) =
∫
C
log |z − λ|µa(dzdz) for
large λ, and this characterizes µa. In particular, the ∗-distribution
of a determines µa – but with a log discontinuity.
![Page 16: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/16.jpg)
Properties of Brown Measure
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
7 / 35
The Brown measure has some nice properties analogous to the
spectral measure, but not all:
• τ(ak) =
∫
C
zk µa(dzdz) and τ(a∗k) =
∫
C
zk µa(dzdz)
but you cannot max and match.
• τ(log |a− λ|) = L(a− λ) =
∫
C
log |z − λ|µa(dzdz) for
large λ, and this characterizes µa. In particular, the ∗-distribution
of a determines µa – but with a log discontinuity.
• suppµa ⊆ Spec(a) (can be a strict subset).
![Page 17: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/17.jpg)
Properties of Brown Measure
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
7 / 35
The Brown measure has some nice properties analogous to the
spectral measure, but not all:
• τ(ak) =
∫
C
zk µa(dzdz) and τ(a∗k) =
∫
C
zk µa(dzdz)
but you cannot max and match.
• τ(log |a− λ|) = L(a− λ) =
∫
C
log |z − λ|µa(dzdz) for
large λ, and this characterizes µa. In particular, the ∗-distribution
of a determines µa – but with a log discontinuity.
• suppµa ⊆ Spec(a) (can be a strict subset).
Let AN be a sequence of matrices with a as limit in ∗-distribution.
Since the Brown measure µAN is the empirical spectral distribution
of AN , it is natural to expect that ESD(AN ) → µa.
![Page 18: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/18.jpg)
Properties of Brown Measure
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
7 / 35
The Brown measure has some nice properties analogous to the
spectral measure, but not all:
• τ(ak) =
∫
C
zk µa(dzdz) and τ(a∗k) =
∫
C
zk µa(dzdz)
but you cannot max and match.
• τ(log |a− λ|) = L(a− λ) =
∫
C
log |z − λ|µa(dzdz) for
large λ, and this characterizes µa. In particular, the ∗-distribution
of a determines µa – but with a log discontinuity.
• suppµa ⊆ Spec(a) (can be a strict subset).
Let AN be a sequence of matrices with a as limit in ∗-distribution.
Since the Brown measure µAN is the empirical spectral distribution
of AN , it is natural to expect that ESD(AN ) → µa. The logdiscontinuity often makes this exceedingly difficult to prove.
![Page 19: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/19.jpg)
Convergence of the Brown Measure
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
8 / 35
Let a, ann∈N be a uniformly bounded set of operators in some
W ∗-probability spaces, with an → a in ∗-distribution. We would
hope that µan → µa. Without some very fine information about the
spectral measure of |an − λ| near the edge of Spec(an), the best
that can be said in general is the following.
![Page 20: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/20.jpg)
Convergence of the Brown Measure
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
8 / 35
Let a, ann∈N be a uniformly bounded set of operators in some
W ∗-probability spaces, with an → a in ∗-distribution. We would
hope that µan → µa. Without some very fine information about the
spectral measure of |an − λ| near the edge of Spec(an), the best
that can be said in general is the following.
Proposition. Suppose that µan → µ weakly for some probability
measure µ on C. Then
∫
C
log |z − λ|µ(dzdz) ≤∫
C
log |z − λ|µa(dzdz)
for all λ ∈ C; and equality holds for sufficiently large λ.
![Page 21: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/21.jpg)
Convergence of the Brown Measure
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
8 / 35
Let a, ann∈N be a uniformly bounded set of operators in some
W ∗-probability spaces, with an → a in ∗-distribution. We would
hope that µan → µa. Without some very fine information about the
spectral measure of |an − λ| near the edge of Spec(an), the best
that can be said in general is the following.
Proposition. Suppose that µan → µ weakly for some probability
measure µ on C. Then
∫
C
log |z − λ|µ(dzdz) ≤∫
C
log |z − λ|µa(dzdz)
for all λ ∈ C; and equality holds for sufficiently large λ.
Corollary. Let Va be the unbounded connected component of
C \ suppµa. Then suppµ ⊆ C \ Va. (In particular, if suppµa is
simply-connected, then suppµ ⊆ suppµa.)
![Page 22: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/22.jpg)
Brown Measure via Regularization
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
9 / 35
The function L(a− λ) =∫
Rlog t µ|a|(dt) is essentially impossible
to compute with. But we can use regularity properties of the spectral
resolution to approach it in a different way. Define
Lǫ(a) =1
2τ(log(a∗a+ ǫ)), ǫ > 0.
![Page 23: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/23.jpg)
Brown Measure via Regularization
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
9 / 35
The function L(a− λ) =∫
Rlog t µ|a|(dt) is essentially impossible
to compute with. But we can use regularity properties of the spectral
resolution to approach it in a different way. Define
Lǫ(a) =1
2τ(log(a∗a+ ǫ)), ǫ > 0.
The function λ 7→ Lǫ(a− λ) is C∞(C), and is subharmonic.
Define
hǫa(λ) =1
2π∇2
λLǫ(a− λ).
Then hǫa is a smooth probability density on C, and
µa(dλ) = limǫ↓0
hǫa(λ) dλ.
![Page 24: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/24.jpg)
Brown Measure via Regularization
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
9 / 35
The function L(a− λ) =∫
Rlog t µ|a|(dt) is essentially impossible
to compute with. But we can use regularity properties of the spectral
resolution to approach it in a different way. Define
Lǫ(a) =1
2τ(log(a∗a+ ǫ)), ǫ > 0.
The function λ 7→ Lǫ(a− λ) is C∞(C), and is subharmonic.
Define
hǫa(λ) =1
2π∇2
λLǫ(a− λ).
Then hǫa is a smooth probability density on C, and
µa(dλ) = limǫ↓0
hǫa(λ) dλ.
It is not difficult to explicitly calculate the density hǫa for fixed ǫ > 0.
![Page 25: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/25.jpg)
The Density hǫa and the Spectrum of a
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
10 / 35
Lemma. Let λ ∈ C, and denote aλ = a− λ. Then
hǫa(λ) =1
πǫτ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)
.
![Page 26: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/26.jpg)
The Density hǫa and the Spectrum of a
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
10 / 35
Lemma. Let λ ∈ C, and denote aλ = a− λ. Then
hǫa(λ) =1
πǫτ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)
.
From here it is easy to see why suppµa ⊆ Spec(a). If λ ∈ Res(a)so that a−1
λ ∈ A, we quickly estimate
∣
∣τ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)∣
∣
≤∥
∥(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
∥
∥
![Page 27: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/27.jpg)
The Density hǫa and the Spectrum of a
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
10 / 35
Lemma. Let λ ∈ C, and denote aλ = a− λ. Then
hǫa(λ) =1
πǫτ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)
.
From here it is easy to see why suppµa ⊆ Spec(a). If λ ∈ Res(a)so that a−1
λ ∈ A, we quickly estimate
∣
∣τ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)∣
∣
≤∥
∥(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
∥
∥
≤∥
∥(a∗λaλ + ǫ)−1∥
∥
∥
∥(aλa∗λ + ǫ)−1
∥
∥
![Page 28: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/28.jpg)
The Density hǫa and the Spectrum of a
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
10 / 35
Lemma. Let λ ∈ C, and denote aλ = a− λ. Then
hǫa(λ) =1
πǫτ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)
.
From here it is easy to see why suppµa ⊆ Spec(a). If λ ∈ Res(a)so that a−1
λ ∈ A, we quickly estimate
∣
∣τ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)∣
∣
≤∥
∥(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
∥
∥
≤∥
∥(a∗λaλ + ǫ)−1∥
∥
∥
∥(aλa∗λ + ǫ)−1
∥
∥
≤∥
∥(a∗λaλ)−1∥
∥
∥
∥(aλa∗λ)
−1∥
∥
![Page 29: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/29.jpg)
The Density hǫa and the Spectrum of a
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
10 / 35
Lemma. Let λ ∈ C, and denote aλ = a− λ. Then
hǫa(λ) =1
πǫτ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)
.
From here it is easy to see why suppµa ⊆ Spec(a). If λ ∈ Res(a)so that a−1
λ ∈ A, we quickly estimate
∣
∣τ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)∣
∣
≤∥
∥(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
∥
∥
≤∥
∥(a∗λaλ + ǫ)−1∥
∥
∥
∥(aλa∗λ + ǫ)−1
∥
∥
≤∥
∥(a∗λaλ)−1∥
∥
∥
∥(aλa∗λ)
−1∥
∥
≤‖(a− λ)−1‖4.
![Page 30: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/30.jpg)
The Density hǫa and the Spectrum of a
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
10 / 35
Lemma. Let λ ∈ C, and denote aλ = a− λ. Then
hǫa(λ) =1
πǫτ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)
.
From here it is easy to see why suppµa ⊆ Spec(a). If λ ∈ Res(a)so that a−1
λ ∈ A, we quickly estimate
∣
∣τ(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)∣
∣
≤∥
∥(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
∥
∥
≤∥
∥(a∗λaλ + ǫ)−1∥
∥
∥
∥(aλa∗λ + ǫ)−1
∥
∥
≤∥
∥(a∗λaλ)−1∥
∥
∥
∥(aλa∗λ)
−1∥
∥
≤‖(a− λ)−1‖4.
This is locally uniformly bounded in λ; so taking ǫ ↓ 0, the factor of ǫin hǫa(λ) kills the term; we find µa = 0 in a neighborhood of λ.
![Page 31: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/31.jpg)
Invertibility in Lp(A, τ)
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
11 / 35
Recall that Lp(A, τ) is the closure of A in the norm
‖a‖pp = τ(|a|p) = τ(
(a∗a)p/2)
.
(It can be realized as a set of densely-defined unbounded operators,
acting on the same Hilbert space as A). The non-commutative
Lp-norms satisfy the same Holder inequality as the classical ones.
![Page 32: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/32.jpg)
Invertibility in Lp(A, τ)
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
11 / 35
Recall that Lp(A, τ) is the closure of A in the norm
‖a‖pp = τ(|a|p) = τ(
(a∗a)p/2)
.
(It can be realized as a set of densely-defined unbounded operators,
acting on the same Hilbert space as A). The non-commutative
Lp-norms satisfy the same Holder inequality as the classical ones.
It is perfectly possible for a ∈ A to be invertible in Lp(A, τ) without
having a bounded inverse. That is: there can exist b ∈ Lp(A, τ) \Awith ab = ba = 1 (viewed as an equation in Lp(A, τ)).
![Page 33: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/33.jpg)
Invertibility in Lp(A, τ)
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
11 / 35
Recall that Lp(A, τ) is the closure of A in the norm
‖a‖pp = τ(|a|p) = τ(
(a∗a)p/2)
.
(It can be realized as a set of densely-defined unbounded operators,
acting on the same Hilbert space as A). The non-commutative
Lp-norms satisfy the same Holder inequality as the classical ones.
It is perfectly possible for a ∈ A to be invertible in Lp(A, τ) without
having a bounded inverse. That is: there can exist b ∈ Lp(A, τ) \Awith ab = ba = 1 (viewed as an equation in Lp(A, τ)).
The preceding proof (with very little change) shows that hǫa(λ) → 0at any point λ where a− λ is invertible in L4(A, τ).
![Page 34: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/34.jpg)
Invertibility in Lp(A, τ)
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
11 / 35
Recall that Lp(A, τ) is the closure of A in the norm
‖a‖pp = τ(|a|p) = τ(
(a∗a)p/2)
.
(It can be realized as a set of densely-defined unbounded operators,
acting on the same Hilbert space as A). The non-commutative
Lp-norms satisfy the same Holder inequality as the classical ones.
It is perfectly possible for a ∈ A to be invertible in Lp(A, τ) without
having a bounded inverse. That is: there can exist b ∈ Lp(A, τ) \Awith ab = ba = 1 (viewed as an equation in Lp(A, τ)).
The preceding proof (with very little change) shows that hǫa(λ) → 0at any point λ where a− λ is invertible in L4(A, τ).
Definition. The Lp(A, τ) resolvent Resp,τ (a) is the interior of the
set of λ ∈ C for which a− λ has an inverse in Lp(A, τ). The
Lp(A, τ) spectrum Specp,τ (a) is C \Resp,τ (a).
![Page 35: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/35.jpg)
The Lp(A, τ) Spectrum
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
12 / 35
From Holder’s inequality, we have the inclusions
Specp,τ (a) ⊆ Specq,τ (a) ⊆ Spec(a)
for 1 ≤ p ≤ q <∞. Without including the closure in the definition,
these inclusions can be strict; with the closure, my (wild) conjecture
is that Spec1,τ (a) = Spec(a) for all a.
![Page 36: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/36.jpg)
The Lp(A, τ) Spectrum
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
12 / 35
From Holder’s inequality, we have the inclusions
Specp,τ (a) ⊆ Specq,τ (a) ⊆ Spec(a)
for 1 ≤ p ≤ q <∞. Without including the closure in the definition,
these inclusions can be strict; with the closure, my (wild) conjecture
is that Spec1,τ (a) = Spec(a) for all a.
As noted, suppµa ⊆ Spec4,τ (a).
![Page 37: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/37.jpg)
The Lp(A, τ) Spectrum
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
12 / 35
From Holder’s inequality, we have the inclusions
Specp,τ (a) ⊆ Specq,τ (a) ⊆ Spec(a)
for 1 ≤ p ≤ q <∞. Without including the closure in the definition,
these inclusions can be strict; with the closure, my (wild) conjecture
is that Spec1,τ (a) = Spec(a) for all a.
As noted, suppµa ⊆ Spec4,τ (a). But we can do better. Recall that
π
ǫhǫa(λ) = τ
(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)
.
If we naıvely set ǫ = 0 on the right-hand-side, we get (heuristically)
τ(
(a∗λaλ)−1(aλa
∗λ)
−1))
= τ(
(a∗λ)−1(aλ)
−2(a∗λ)−1)
![Page 38: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/38.jpg)
The Lp(A, τ) Spectrum
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
12 / 35
From Holder’s inequality, we have the inclusions
Specp,τ (a) ⊆ Specq,τ (a) ⊆ Spec(a)
for 1 ≤ p ≤ q <∞. Without including the closure in the definition,
these inclusions can be strict; with the closure, my (wild) conjecture
is that Spec1,τ (a) = Spec(a) for all a.
As noted, suppµa ⊆ Spec4,τ (a). But we can do better. Recall that
π
ǫhǫa(λ) = τ
(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)
.
If we naıvely set ǫ = 0 on the right-hand-side, we get (heuristically)
τ(
(a∗λaλ)−1(aλa
∗λ)
−1))
= τ(
(a∗λ)−1(aλ)
−2(a∗λ)−1)
= τ(
(a−2λ )∗a−2
λ
)
= ‖a−2λ ‖22.
![Page 39: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/39.jpg)
The Lp(A, τ) Spectrum
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
12 / 35
From Holder’s inequality, we have the inclusions
Specp,τ (a) ⊆ Specq,τ (a) ⊆ Spec(a)
for 1 ≤ p ≤ q <∞. Without including the closure in the definition,
these inclusions can be strict; with the closure, my (wild) conjecture
is that Spec1,τ (a) = Spec(a) for all a.
As noted, suppµa ⊆ Spec4,τ (a). But we can do better. Recall that
π
ǫhǫa(λ) = τ
(
(a∗λaλ + ǫ)−1(aλa∗λ + ǫ)−1
)
.
If we naıvely set ǫ = 0 on the right-hand-side, we get (heuristically)
τ(
(a∗λaλ)−1(aλa
∗λ)
−1))
= τ(
(a∗λ)−1(aλ)
−2(a∗λ)−1)
= τ(
(a−2λ )∗a−2
λ
)
= ‖a−2λ ‖22.
Note, this is not equal to ‖a−1λ ‖44 when aλ is not normal.
![Page 40: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/40.jpg)
The L22,τ Spectrum
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
13 / 35
Proposition. Let a ∈ A, and suppose a2 is invertible in L2(A, τ).Then for all ǫ > 0,
τ(
(a∗a+ ǫ)−1(aa∗ + ǫ)−1)
≤ ‖a−2‖22.
(The proof is trickier than you might think.)
![Page 41: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/41.jpg)
The L22,τ Spectrum
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
13 / 35
Proposition. Let a ∈ A, and suppose a2 is invertible in L2(A, τ).Then for all ǫ > 0,
τ(
(a∗a+ ǫ)−1(aa∗ + ǫ)−1)
≤ ‖a−2‖22.
(The proof is trickier than you might think.)
Definition. The L22,τ resolvent of a, Res22,τ (a), is the interior of the
set of λ ∈ C for which (a− λ)2 is invertible in L2(A, τ). The L22,τ
spectrum of a is Spec22,τ (a) = C \Res22,τ (a).
![Page 42: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/42.jpg)
The L22,τ Spectrum
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
13 / 35
Proposition. Let a ∈ A, and suppose a2 is invertible in L2(A, τ).Then for all ǫ > 0,
τ(
(a∗a+ ǫ)−1(aa∗ + ǫ)−1)
≤ ‖a−2‖22.
(The proof is trickier than you might think.)
Definition. The L22,τ resolvent of a, Res22,τ (a), is the interior of the
set of λ ∈ C for which (a− λ)2 is invertible in L2(A, τ). The L22,τ
spectrum of a is Spec22,τ (a) = C \Res22,τ (a).
Theorem. suppµa ⊆ Spec22,τ (a).
![Page 43: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/43.jpg)
The L22,τ Spectrum
• Dedication
• Citations
Brown Measure
• Brown Measure
• Circular
• Properties
• Convergence
• Regularize
• Spectrum
•Lp
Inverse
•Lp
Spectrum
• Support
Brownian Motion
Segal–Bargmann
Brown Measure Support
13 / 35
Proposition. Let a ∈ A, and suppose a2 is invertible in L2(A, τ).Then for all ǫ > 0,
τ(
(a∗a+ ǫ)−1(aa∗ + ǫ)−1)
≤ ‖a−2‖22.
(The proof is trickier than you might think.)
Definition. The L22,τ resolvent of a, Res22,τ (a), is the interior of the
set of λ ∈ C for which (a− λ)2 is invertible in L2(A, τ). The L22,τ
spectrum of a is Spec22,τ (a) = C \Res22,τ (a).
Theorem. suppµa ⊆ Spec22,τ (a).
Another wild conjecture: this is actually equality. (That depends on
showing that, if a2 is not invertible in L2(A, τ), the above quantity
blows up at rate Ω(1/ǫ). This appears to be what happens in the
case that a is normal, which would imply Spec22,τ (a) = Spec4,τ (a)= Spec(a) in that case.)
![Page 44: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/44.jpg)
Brownian Motion on U(N),GL(N,C), and the Large-N Limit
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
14 / 35
![Page 45: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/45.jpg)
Brownian Motion on Lie Groups
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
15 / 35
On any Riemannian manifold M , there’s a Laplace operator ∆M .
And where there’s a Laplacian, there’s a Brownian motion: the
Markov process (Bxt )t≥0 on M with generator 1
2∆M , started at
Bx0 = x ∈M .
![Page 46: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/46.jpg)
Brownian Motion on Lie Groups
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
15 / 35
On any Riemannian manifold M , there’s a Laplace operator ∆M .
And where there’s a Laplacian, there’s a Brownian motion: the
Markov process (Bxt )t≥0 on M with generator 1
2∆M , started at
Bx0 = x ∈M .
Let Γ be a (matrix) Lie group. Any inner product on Lie(Γ) = TIΓgives rise to a unique left-invariant Riemannian metric, and
corresponding Laplacian ∆Γ. On Γ we canonically start the
Brownian motion (Bt)t≥0 at I ∈ Γ.
![Page 47: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/47.jpg)
Brownian Motion on Lie Groups
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
15 / 35
On any Riemannian manifold M , there’s a Laplace operator ∆M .
And where there’s a Laplacian, there’s a Brownian motion: the
Markov process (Bxt )t≥0 on M with generator 1
2∆M , started at
Bx0 = x ∈M .
Let Γ be a (matrix) Lie group. Any inner product on Lie(Γ) = TIΓgives rise to a unique left-invariant Riemannian metric, and
corresponding Laplacian ∆Γ. On Γ we canonically start the
Brownian motion (Bt)t≥0 at I ∈ Γ.
There is a beautiful relationship between the Brownian motion Wt on
the Lie algebra Lie(Γ) and the Brownian motion Bt: the rolling map
dBt = Bt dWt, i.e. Bt = I +
∫ t
0Bt dWt.
Here denotes the Stratonovich stochastic integral. This can always
be converted into an Ito integral; but the answer depends on the
structure of the group Γ (and the chosen inner product).
![Page 48: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/48.jpg)
Brownian Motion on U(N) and GL(N,C)
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
16 / 35
Fix the reverse normalized Hilbert–Schmidt inner product on
MN (C) for all matrix Lie algebras:
〈A,B〉 = NTr(B∗A).
Let Xt = XNt and Yt = Y N
t be independent Hermitian Brownian
motions of variance t/N .
![Page 49: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/49.jpg)
Brownian Motion on U(N) and GL(N,C)
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
16 / 35
Fix the reverse normalized Hilbert–Schmidt inner product on
MN (C) for all matrix Lie algebras:
〈A,B〉 = NTr(B∗A).
Let Xt = XNt and Yt = Y N
t be independent Hermitian Brownian
motions of variance t/N .
The Brownian motion on Lie(U(N)) is iXt; the Brownian motion
Ut on U(N) satisfies
dUt = iUt dXt − 12Ut dt.
![Page 50: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/50.jpg)
Brownian Motion on U(N) and GL(N,C)
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
16 / 35
Fix the reverse normalized Hilbert–Schmidt inner product on
MN (C) for all matrix Lie algebras:
〈A,B〉 = NTr(B∗A).
Let Xt = XNt and Yt = Y N
t be independent Hermitian Brownian
motions of variance t/N .
The Brownian motion on Lie(U(N)) is iXt; the Brownian motion
Ut on U(N) satisfies
dUt = iUt dXt − 12Ut dt.
The Brownian motion on Lie(GL(N,C)) = MN (C) is
Zt = 2−1/2i(Xt + iYt); the Brownian motion Gt on GL(N,C)satisfies
dGt = Gt dZt.
![Page 51: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/51.jpg)
Free Additive Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
17 / 35
If Xt = XNt is a Hermitian Brownian motion process, then at each
time t > 0 it is a GUEN with entries of variance t/N . Wigner’s law
then shows that the empirical spectral distribution of XNt converges
to the semicircle law ςt =1
2πt
√
(4t− x2)+ dx.
1 1
![Page 52: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/52.jpg)
Free Additive Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
17 / 35
If Xt = XNt is a Hermitian Brownian motion process, then at each
time t > 0 it is a GUEN with entries of variance t/N . Wigner’s law
then shows that the empirical spectral distribution of XNt converges
to the semicircle law ςt =1
2πt
√
(4t− x2)+ dx. In fact, it converges
as a process.
1 1
![Page 53: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/53.jpg)
Free Additive Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
17 / 35
If Xt = XNt is a Hermitian Brownian motion process, then at each
time t > 0 it is a GUEN with entries of variance t/N . Wigner’s law
then shows that the empirical spectral distribution of XNt converges
to the semicircle law ςt =1
2πt
√
(4t− x2)+ dx. In fact, it converges
as a process.
A process (xt)t≥0 (in a W ∗-probability space with trace τ ) is a free
additive Brownian motion if its increments are freely independent
— xt − xs is free from xr : r ≤ s — and xt − xs has the
semicircular distribution ςt−s, for all t > s.
1 1
![Page 54: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/54.jpg)
Free Additive Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
17 / 35
If Xt = XNt is a Hermitian Brownian motion process, then at each
time t > 0 it is a GUEN with entries of variance t/N . Wigner’s law
then shows that the empirical spectral distribution of XNt converges
to the semicircle law ςt =1
2πt
√
(4t− x2)+ dx. In fact, it converges
as a process.
A process (xt)t≥0 (in a W ∗-probability space with trace τ ) is a free
additive Brownian motion if its increments are freely independent
— xt − xs is free from xr : r ≤ s — and xt − xs has the
semicircular distribution ςt−s, for all t > s. It can be constructed on
the free Fock space over L2(R+): xt = l(1[0,t]) + l∗(1[0,t]).
![Page 55: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/55.jpg)
Free Additive Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
17 / 35
If Xt = XNt is a Hermitian Brownian motion process, then at each
time t > 0 it is a GUEN with entries of variance t/N . Wigner’s law
then shows that the empirical spectral distribution of XNt converges
to the semicircle law ςt =1
2πt
√
(4t− x2)+ dx. In fact, it converges
as a process.
A process (xt)t≥0 (in a W ∗-probability space with trace τ ) is a free
additive Brownian motion if its increments are freely independent
— xt − xs is free from xr : r ≤ s — and xt − xs has the
semicircular distribution ςt−s, for all t > s. It can be constructed on
the free Fock space over L2(R+): xt = l(1[0,t]) + l∗(1[0,t]).
In 1991, Voiculescu showed that the Hermitian Brownian motion
(XNt )t≥0 converges to (xt)t≥0 in finite-dimensional
non-commutative distributions:
1
NTr(P (Xt1 , . . . , Xtn)) → τ(P (xt1 , . . . , xtn)) ∀P.
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Free Unitary and Free Multiplicative Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
18 / 35
There is now a well-developed theory of free stochastic differential
equations. Initially constructed in the free Fock space setting (by
Kummerer and Speicher in the early 1990s), it was used by Biane in
1997 to define “free versions” of Ut and Gt.
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Free Unitary and Free Multiplicative Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
18 / 35
There is now a well-developed theory of free stochastic differential
equations. Initially constructed in the free Fock space setting (by
Kummerer and Speicher in the early 1990s), it was used by Biane in
1997 to define “free versions” of Ut and Gt.
Let xt, yt be freely independent free additive Brownian motions, and
zt = 2−1/2i(xt + iyt). The free unitary Brownian motion is the
process started at u0 = 1 defined by
dut = iut dxt − 12ut dt.
The free multiplicative Brownian motion is the process started at
g0 = 1 defined by
dgt = gt dzt.
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Free Unitary and Free Multiplicative Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
18 / 35
There is now a well-developed theory of free stochastic differential
equations. Initially constructed in the free Fock space setting (by
Kummerer and Speicher in the early 1990s), it was used by Biane in
1997 to define “free versions” of Ut and Gt.
Let xt, yt be freely independent free additive Brownian motions, and
zt = 2−1/2i(xt + iyt). The free unitary Brownian motion is the
process started at u0 = 1 defined by
dut = iut dxt − 12ut dt.
The free multiplicative Brownian motion is the process started at
g0 = 1 defined by
dgt = gt dzt.
It is natural to expect that these processes should be the large-Nlimits of the U(N) and GL(N,C) Brownian motions.
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Free Unitary Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
19 / 35
Theorem. [Biane, 1997] For all non-commutative (Laurent)
polynomials P in n variables and times t1, . . . , tn ≥ 0,
1
NTr(P (UN
t1 , . . . , UNtn )) → τ(P (ut1 , . . . , utn)) a.s.
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Free Unitary Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
19 / 35
Theorem. [Biane, 1997] For all non-commutative (Laurent)
polynomials P in n variables and times t1, . . . , tn ≥ 0,
1
NTr(P (UN
t1 , . . . , UNtn )) → τ(P (ut1 , . . . , utn)) a.s.
Biane also computed the moments of ut, and its spectral measure
νt: it has a density (smooth on the interior of its support), supported
on a compact arc for t < 4, and fully supported on U for t ≥ 4.
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Free Unitary Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
19 / 35
Theorem. [Biane, 1997] For all non-commutative (Laurent)
polynomials P in n variables and times t1, . . . , tn ≥ 0,
1
NTr(P (UN
t1 , . . . , UNtn )) → τ(P (ut1 , . . . , utn)) a.s.
Biane also computed the moments of ut, and its spectral measure
νt: it has a density (smooth on the interior of its support), supported
on a compact arc for t < 4, and fully supported on U for t ≥ 4.
-3 -2 -1 0 1 2 30
100
200
300
400
500
600
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Analytic Transforms Related to ut
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
20 / 35
Biane’s approach to understanding the measure νt was through its
moment-generating function
ψt(z) =
∫
U
uz
1− uzνt(du) =
∑
n≥1
mn(νt) zn
(the second = holds for |z| < 1; the integral converges for
1/z /∈ supp νt).
![Page 63: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/63.jpg)
Analytic Transforms Related to ut
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
20 / 35
Biane’s approach to understanding the measure νt was through its
moment-generating function
ψt(z) =
∫
U
uz
1− uzνt(du) =
∑
n≥1
mn(νt) zn
(the second = holds for |z| < 1; the integral converges for
1/z /∈ supp νt). Then define
χt(z) =ψt(z)
1 + ψt(z).
The function χt is injective on D, and has a one-sided inverse ft:ft(χt(z)) = z for z ∈ D (but χt ft is only the identity on a certain
region in C; more on this later).
![Page 64: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/64.jpg)
Analytic Transforms Related to ut
• Dedication
• Citations
Brown Measure
Brownian Motion
• BM on Lie Groups
•U & GL
• Free+BM
• Free×BM
• Free Unitary BM
• Transforms
• Free Mult. BM
•GL Spectrum
Segal–Bargmann
Brown Measure Support
20 / 35
Biane’s approach to understanding the measure νt was through its
moment-generating function
ψt(z) =
∫
U
uz
1− uzνt(du) =
∑
n≥1
mn(νt) zn
(the second = holds for |z| < 1; the integral converges for
1/z /∈ supp νt). Then define
χt(z) =ψt(z)
1 + ψt(z).
The function χt is injective on D, and has a one-sided inverse ft:ft(χt(z)) = z for z ∈ D (but χt ft is only the identity on a certain
region in C; more on this later).
Using the SDE for ut and some clever complex analysis, Biane
showed that
ft(z) = zet
21+z
1−z .
![Page 65: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/65.jpg)
The Large-N Limit of GNt
21 / 35
In 1997 Biane conjectured a similar large-N limit should hold for the Brownian
motion on GL(N,C), but the ideas of his UNt proof (spectral theorem,
representation theory of U(N)) did not translate well to the a.s. non-normal
process GNt .
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The Large-N Limit of GNt
21 / 35
In 1997 Biane conjectured a similar large-N limit should hold for the Brownian
motion on GL(N,C), but the ideas of his UNt proof (spectral theorem,
representation theory of U(N)) did not translate well to the a.s. non-normal
process GNt .
Theorem. [K, 2014 (2016)] For all non-commutative Laurent polynomials P in
2n variables, and times t1, . . . , tn ≥ 0,
1
NTr(
P (GNt1 , (G
Nt1 )
∗, . . . , GNtn , (G
Ntn)
∗))
→ τ(
P (gt1 , g∗t1 , . . . , gtn , g
∗tn))
a.s.
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The Large-N Limit of GNt
21 / 35
In 1997 Biane conjectured a similar large-N limit should hold for the Brownian
motion on GL(N,C), but the ideas of his UNt proof (spectral theorem,
representation theory of U(N)) did not translate well to the a.s. non-normal
process GNt .
Theorem. [K, 2014 (2016)] For all non-commutative Laurent polynomials P in
2n variables, and times t1, . . . , tn ≥ 0,
1
NTr(
P (GNt1 , (G
Nt1 )
∗, . . . , GNtn , (G
Ntn)
∗))
→ τ(
P (gt1 , g∗t1 , . . . , gtn , g
∗tn))
a.s.
The proof required several new ingredients: a detailed understanding of the
Laplacian on GL(N,C), and concentration of measure for trace polynomials.
Putting these together with an iteration scheme from the SDE, together with
requisite covariance estimates, yielded the proof.
![Page 68: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/68.jpg)
The Large-N Limit of GNt
21 / 35
In 1997 Biane conjectured a similar large-N limit should hold for the Brownian
motion on GL(N,C), but the ideas of his UNt proof (spectral theorem,
representation theory of U(N)) did not translate well to the a.s. non-normal
process GNt .
Theorem. [K, 2014 (2016)] For all non-commutative Laurent polynomials P in
2n variables, and times t1, . . . , tn ≥ 0,
1
NTr(
P (GNt1 , (G
Nt1 )
∗, . . . , GNtn , (G
Ntn)
∗))
→ τ(
P (gt1 , g∗t1 , . . . , gtn , g
∗tn))
a.s.
The proof required several new ingredients: a detailed understanding of the
Laplacian on GL(N,C), and concentration of measure for trace polynomials.
Putting these together with an iteration scheme from the SDE, together with
requisite covariance estimates, yielded the proof.
This is convergence of the (multi-time) ∗-distribution, of a non-normal matrix
process. What about the eigenvalues?
![Page 69: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/69.jpg)
The Eigenvalues of Brownian Motion GL(N,C)
22 / 35
Because UNt and ut are normal, their ∗-distributions encode their ESDs, so
the bulk eigenvalue behavior is fully understood.
![Page 70: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/70.jpg)
The Eigenvalues of Brownian Motion GL(N,C)
22 / 35
Because UNt and ut are normal, their ∗-distributions encode their ESDs, so
the bulk eigenvalue behavior is fully understood.
The GL(N,C) Brownian motion GNt eigenvalues are much more challenging.
![Page 71: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/71.jpg)
The Eigenvalues of Brownian Motion GL(N,C)
22 / 35
Because UNt and ut are normal, their ∗-distributions encode their ESDs, so
the bulk eigenvalue behavior is fully understood.
The GL(N,C) Brownian motion GNt eigenvalues are much more challenging.
t = 1
![Page 72: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/72.jpg)
The Eigenvalues of Brownian Motion GL(N,C)
22 / 35
Because UNt and ut are normal, their ∗-distributions encode their ESDs, so
the bulk eigenvalue behavior is fully understood.
The GL(N,C) Brownian motion GNt eigenvalues are much more challenging.
t = 2
![Page 73: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/73.jpg)
The Eigenvalues of Brownian Motion GL(N,C)
22 / 35
Because UNt and ut are normal, their ∗-distributions encode their ESDs, so
the bulk eigenvalue behavior is fully understood.
The GL(N,C) Brownian motion GNt eigenvalues are much more challenging.
t = 4
![Page 74: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/74.jpg)
The Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
23 / 35
![Page 75: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/75.jpg)
The Unitary Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
24 / 35
The Segal–Bargmann (Hall) Transform is a map from functions on
U(N) to holomorphic functions on GL(N,C). It is defined by the
analytic continuation of the action of the heat operator:
BNt f =
(
et
2∆U(N)f
)
C
.
![Page 76: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/76.jpg)
The Unitary Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
24 / 35
The Segal–Bargmann (Hall) Transform is a map from functions on
U(N) to holomorphic functions on GL(N,C). It is defined by the
analytic continuation of the action of the heat operator:
BNt f =
(
et
2∆U(N)f
)
C
.
Writing out what this integral formula means in probabilistic terms,
here is a nice way to express it: let F already be a holomorphic
function on GL(N),C), and let f = F |U(N). Let Ut and Gt be
independent Brownian motions on U(N) and GL(N,C). Then
(Btf)(Gt) = E[F (GtUt)|Gt].
![Page 77: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/77.jpg)
The Unitary Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
24 / 35
The Segal–Bargmann (Hall) Transform is a map from functions on
U(N) to holomorphic functions on GL(N,C). It is defined by the
analytic continuation of the action of the heat operator:
BNt f =
(
et
2∆U(N)f
)
C
.
Writing out what this integral formula means in probabilistic terms,
here is a nice way to express it: let F already be a holomorphic
function on GL(N),C), and let f = F |U(N). Let Ut and Gt be
independent Brownian motions on U(N) and GL(N,C). Then
(Btf)(Gt) = E[F (GtUt)|Gt].
This extends beyond f that already possess an analytic
continuation; it defines an isometric isomorphism
BNt : L2(U(N), Ut) → HL2(GL(N,C), Gt).
![Page 78: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/78.jpg)
The Free Unitary Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
25 / 35
In 1997, Biane introduced a free version of the Unitary SBT, which
can be described in similar terms: acting on, say, polynomials f in a
single variable, Gtf is defined by
(Gtf)(gt) = τ [f(gtut)|gt].
He conjectured that Gt is the large-N limit of BNt in an appropriate
sense; this was proven by Driver, Hall, and me in 2013. (It was for
this work that we invented trace polynomial concentration.)
![Page 79: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/79.jpg)
The Free Unitary Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
25 / 35
In 1997, Biane introduced a free version of the Unitary SBT, which
can be described in similar terms: acting on, say, polynomials f in a
single variable, Gtf is defined by
(Gtf)(gt) = τ [f(gtut)|gt].
He conjectured that Gt is the large-N limit of BNt in an appropriate
sense; this was proven by Driver, Hall, and me in 2013. (It was for
this work that we invented trace polynomial concentration.)
Biane proved directly (and it follows from the large-N limit) that Gt
extends to an isometric isomorphism
Gt : L2(U, νt) → At
where At is a certain reproducing-kernel Hilbert space of
holomorphic functions. The norm on At is given by
‖F‖2At= τ(|F (gt)|2) = τ(F (gt)
∗F (gt)) = ‖F (gt)‖22.
![Page 80: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/80.jpg)
The Range of the Free Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
26 / 35
The functions F ∈ At are not all entire functions. They are
holomorphic on a bounded region Σt
Σt = C \ χt(C \ supp νt)
where (recall) χt is the (right-)inverse of ft(z) = zet
21+z
1−z .
![Page 81: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/81.jpg)
The Range of the Free Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
26 / 35
0 5 10
-5
0
5
t = 1
![Page 82: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/82.jpg)
The Range of the Free Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
26 / 35
0 5 10
-5
0
5
t = 2
![Page 83: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/83.jpg)
The Range of the Free Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
26 / 35
0 5 10
-5
0
5
t = 3.9
![Page 84: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/84.jpg)
The Range of the Free Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
26 / 35
0 5 10
-5
0
5
t = 4
![Page 85: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/85.jpg)
The Range of the Free Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
26 / 35
t = 4
![Page 86: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/86.jpg)
The Range of the Free Segal–Bargmann Transform
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
• SBT
• Free SBT
•Σt
Brown Measure Support
26 / 35
-5 0 5 10
-5
0
5
t = 4.01
![Page 87: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/87.jpg)
The Brown Measure of Free
Multiplicative Brownian Motion
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
27 / 35
![Page 88: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/88.jpg)
The Support of The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
28 / 35
Theorem. (Hall, K, 2018)
suppµgt ⊆ Σt.
![Page 89: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/89.jpg)
The Support of The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
28 / 35
Theorem. (Hall, K, 2018)
suppµgt ⊆ Σt.
Proof. We show that Spec22,τ (gt) = Σt. Equivalently, from the
definition of Σt, we show that Res22,τ (gt) = χt(C \ supp νt).
![Page 90: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/90.jpg)
The Support of The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
28 / 35
Theorem. (Hall, K, 2018)
suppµgt ⊆ Σt.
Proof. We show that Spec22,τ (gt) = Σt. Equivalently, from the
definition of Σt, we show that Res22,τ (gt) = χt(C \ supp νt).
Essentially, λ ∈ Res22,τ (gt) iff (gt − λ)2 is invertible in L2(τ), i.e.
∞ > τ(
|(gt − λ)−2|2)
![Page 91: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/91.jpg)
The Support of The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
28 / 35
Theorem. (Hall, K, 2018)
suppµgt ⊆ Σt.
Proof. We show that Spec22,τ (gt) = Σt. Equivalently, from the
definition of Σt, we show that Res22,τ (gt) = χt(C \ supp νt).
Essentially, λ ∈ Res22,τ (gt) iff (gt − λ)2 is invertible in L2(τ), i.e.
∞ > τ(
|(gt − λ)−2|2)
= ‖(z − λ)−2‖2At.
![Page 92: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/92.jpg)
The Support of The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
28 / 35
Theorem. (Hall, K, 2018)
suppµgt ⊆ Σt.
Proof. We show that Spec22,τ (gt) = Σt. Equivalently, from the
definition of Σt, we show that Res22,τ (gt) = χt(C \ supp νt).
Essentially, λ ∈ Res22,τ (gt) iff (gt − λ)2 is invertible in L2(τ), i.e.
∞ > τ(
|(gt − λ)−2|2)
= ‖(z − λ)−2‖2At.
Recall that Gt is an isometry from L2(U, νt) onto At. Can we find a
function αλt on U with Gt(α
λt )(z) = (z − λ)−2?
![Page 93: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/93.jpg)
The Support of The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
28 / 35
Theorem. (Hall, K, 2018)
suppµgt ⊆ Σt.
Proof. We show that Spec22,τ (gt) = Σt. Equivalently, from the
definition of Σt, we show that Res22,τ (gt) = χt(C \ supp νt).
Essentially, λ ∈ Res22,τ (gt) iff (gt − λ)2 is invertible in L2(τ), i.e.
∞ > τ(
|(gt − λ)−2|2)
= ‖(z − λ)−2‖2At.
Recall that Gt is an isometry from L2(U, νt) onto At. Can we find a
function αλt on U with Gt(α
λt )(z) = (z − λ)−2?
Using PDE techniques, we can compute that
G−1t ((z − λ)−1) =
1
λ
ft(λ)
ft(λ)− u.
![Page 94: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/94.jpg)
The Support of The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
29 / 35
Gt :1
λ
ft(λ)
ft(λ)− u7→ 1
z − λ.
Since 1(z−λ)2
= ddλ
1z−λ , using regularity properties of Gt we have
αλt (u) =
d
dλ
(
1
λ
ft(λ)
ft(λ)− u
)
.
![Page 95: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/95.jpg)
The Support of The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
29 / 35
Gt :1
λ
ft(λ)
ft(λ)− u7→ 1
z − λ.
Since 1(z−λ)2
= ddλ
1z−λ , using regularity properties of Gt we have
αλt (u) =
d
dλ
(
1
λ
ft(λ)
ft(λ)− u
)
.
The question is: for which λ is αλt ∈ L2(U, νt)? I.e.
∫
U
|αλt (u)|2 νt(du) <∞.
![Page 96: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/96.jpg)
The Support of The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
29 / 35
Gt :1
λ
ft(λ)
ft(λ)− u7→ 1
z − λ.
Since 1(z−λ)2
= ddλ
1z−λ , using regularity properties of Gt we have
αλt (u) =
d
dλ
(
1
λ
ft(λ)
ft(λ)− u
)
.
The question is: for which λ is αλt ∈ L2(U, νt)? I.e.
∫
U
|αλt (u)|2 νt(du) <∞.
The answer is: precisely when ft(λ) /∈ supp νt. I.e.
Res22,τ (gt) = f−1t (C \ supp νt) = χt(C \ supp νt).
![Page 97: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/97.jpg)
The Empirical Spectrum and Σt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
30 / 35
Here is a simulation of eigenvalues of G(N)t for N = 2000, together
with the boundary of Σt, at t = 3 (produced in Mathematica).
-2 0 2 4 6 8
-4
-2
0
2
4
![Page 98: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/98.jpg)
The Empirical Spectrum and Σt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
31 / 35
N = 2000, t = 2, 3.9, 4, 4.1.
![Page 99: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/99.jpg)
Computing the Brown Measure
32 / 35
Very recently, jointly with Driver and Hall, we have been able to push further
and actually compute the Brown measure.
![Page 100: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/100.jpg)
Computing the Brown Measure
32 / 35
Very recently, jointly with Driver and Hall, we have been able to push further
and actually compute the Brown measure. To describe it, we need an auxiliary
implicit function = (t, θ), determined by
1− cos θ√
1− 2log
(
2− 2 + 2√
1− 2
2
)
= t.
This defines a real analytic function for |θ| < θmax(t) = cos−1(1− t/2) ∧ π,
which is precisely the argument range of Σt:
![Page 101: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/101.jpg)
The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
33 / 35
Theorem. (Driver, Hall, K, 2018)
suppµgt = Σt.
1
![Page 102: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/102.jpg)
The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
33 / 35
Theorem. (Driver, Hall, K, 2018)
suppµgt = Σt. Moreover, µgt has a continuous density on Σt, that
is real analytic and strictly positive on Σt.
1
![Page 103: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/103.jpg)
The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
33 / 35
Theorem. (Driver, Hall, K, 2018)
suppµgt = Σt. Moreover, µgt has a continuous density on Σt, that
is real analytic and strictly positive on Σt. The density has the form
dµgt =1
r2wt(e
iθ)1Σtrdrdθ
for the real analytic function wt : U → R+ given by
wt(eiθ) =
1
4π
(
2
t+
∂
∂θ
(t, θ) sin θ
1− (t, θ) cos θ
)
.
![Page 104: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/104.jpg)
The Brown Measure of gt
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
33 / 35
Theorem. (Driver, Hall, K, 2018)
suppµgt = Σt. Moreover, µgt has a continuous density on Σt, that
is real analytic and strictly positive on Σt. The density has the form
dµgt =1
r2wt(e
iθ)1Σtrdrdθ
for the real analytic function wt : U → R+ given by
wt(eiθ) =
1
4π
(
2
t+
∂
∂θ
(t, θ) sin θ
1− (t, θ) cos θ
)
.
The techniques needed to prove this theorem are wholly disjoint
from the concepts discussed in this talk so far; they rely primarily on
PDE methods. You’ll have to wait to see those ideas until the next
meeting (Oberwolfach, or Montreal).
![Page 105: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/105.jpg)
Histogram of Eigenvalue Arguments
34 / 35
Here are histograms of complex arguments of eigenvalues of GNt , together
with the argument density of µgt , for N = 2000 and t = 2, 3.8, 4, 5.
![Page 106: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/106.jpg)
Remaining Questions
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
35 / 35
• Explore relations between the Lp(τ)-spectra, in general. They
are probably all equal to the spectrum for gt.
![Page 107: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/107.jpg)
Remaining Questions
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
35 / 35
• Explore relations between the Lp(τ)-spectra, in general. They
are probably all equal to the spectrum for gt.
• Prove that the ESD of GNt actually converges to µgt . (What we
can now say definitively is that the limit ESD is supported in Σt.)
![Page 108: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/108.jpg)
Remaining Questions
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
35 / 35
• Explore relations between the Lp(τ)-spectra, in general. They
are probably all equal to the spectrum for gt.
• Prove that the ESD of GNt actually converges to µgt . (What we
can now say definitively is that the limit ESD is supported in Σt.)
• There is a two-parameter family of invariant diffusions on
GL(N,C) that includes UNt and GN
t , all of which have large-Nlimits described by free SDEs. How much of all this extends to
the whole family? (Our preprint already covers the support in the
two-parameter setting; the density is yet unknown.)
![Page 109: Brown’s Spectral Measure, and the Free Multiplicative ...fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Kemp.pdfBrown Measure Brownian Motion Segal–Bargmann Brown Measure Support](https://reader033.vdocument.in/reader033/viewer/2022060507/5f1f8b8a29340d0d50249002/html5/thumbnails/109.jpg)
Remaining Questions
• Dedication
• Citations
Brown Measure
Brownian Motion
Segal–Bargmann
Brown Measure Support
• Main Theorem
• Proof
• Simulations
• Simulations
• Implicit
• The Brown Measure
• Simulations
• Questions
35 / 35
• Explore relations between the Lp(τ)-spectra, in general. They
are probably all equal to the spectrum for gt.
• Prove that the ESD of GNt actually converges to µgt . (What we
can now say definitively is that the limit ESD is supported in Σt.)
• There is a two-parameter family of invariant diffusions on
GL(N,C) that includes UNt and GN
t , all of which have large-Nlimits described by free SDEs. How much of all this extends to
the whole family? (Our preprint already covers the support in the
two-parameter setting; the density is yet unknown.)
I’ll let you know what more I know next time we meet. ©