![Page 1: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/1.jpg)
MARINUS A. KAASHOEK:
half a century of operator theory
in
Amsterdam
Opening Lecture IWOTA 2017 (Chemnitz)
by
Harm BartErasmus University Rotterdam
1
![Page 2: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/2.jpg)
Born 1937
Ridderkerk
2
![Page 3: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/3.jpg)
Studied in Leiden
Oldest University in The Netherlands
Founded in 1575
3
![Page 4: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/4.jpg)
PhD in 1964 (Leiden)
Postdoc University of California at Los Angeles, 1965 - 1966
VU University Amsterdam 1966 – 2002
Emeritus Professor 2002 – · · · (active!)
4
![Page 5: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/5.jpg)
Hence the title:
half a century of operator theory
in
Amsterdam
5
![Page 6: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/6.jpg)
Many Ph.D students (17)
MatScinet: 234 publications
Co-author of 9 books
Lots of collaborators
6
![Page 7: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/7.jpg)
Research interests mentioned in CV:
Analysis and Operator Theory,
and various connections between Operator Theory Matrix The-
ory and Mathematical Systems Theory
In particular, Wiener-Hopf integral equations and Toeplitz
operators and their nonstationary variants
State space methods for problems in Analysis
Metric constrained interpolation problems,
and various extension and completion problems for
partially defined matrices or operators,
including relaxed commutant lifting problems.
7
![Page 8: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/8.jpg)
In the available time impossible to cover
all aspects
all connections
all references
Aim:Just to give an impression on what Kaashoek has been workingon
Emphasis on ideas / less on specific results
Not all the time mentioning of co-authors involvedList of them (MathSciNet):
8
![Page 9: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/9.jpg)
9
![Page 10: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/10.jpg)
PhD in 1964
Supervisor A.C. Zaanen
10
![Page 11: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/11.jpg)
Doctoral Thesis:
Closed linear operators on Banach spaces
One of the issues:
Local behavior of operator pencils λS − T
11
![Page 12: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/12.jpg)
Sufficient conditions for
dim Ker (λS − T ), codim Im (λS − T )
to be constant on deleted neighborhood of the origin
Determination size of the neighborhood
Extension work of Gohberg/Krein and Kato
12
![Page 13: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/13.jpg)
Postdoc University of California at Los Angeles, 1965 - 1966
Upon return to Amsterdam:
Suggestion to HB: try generalization
pencils λS − T → analytic operator functions W (λ)
(admitting local power series expansions)
13
![Page 14: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/14.jpg)
Important tool: linearization / reduction to the pencil case:
local properties W (λ)↔ local properties λSW −TW
Suitable operators SW and TW on ’big(ger)’ spaces
Defined in terms of power series expansion
W (λ) =∞∑k=0
λkWk
Inspired by (among others, but mainly)
Karl-Heinz Foerster († 29–1–2017)
14
![Page 15: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/15.jpg)
15
![Page 16: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/16.jpg)
Local properties W (λ)↔ local properties pencil λSW −TW
Drawbacks:
local behavior instead of global behavior
pencil λSW −TW instead of spectral item λIW −TW
Helpful in some circumstances: SW left invertible
1975: Enters Israel Gohberg
16
![Page 17: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/17.jpg)
17
![Page 18: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/18.jpg)
Gohberg, Kaashoek, Lay:
Global reduction to spectral case λI−T
(killing two birds with one stone)
Linearization by equivalence after extension
[W (λ) 0
0 IZ
]= E(λ)(λI−TW )F (λ)
E(λ), F (λ) analytic equivalence functions
(Many) properties W (λ)↔ properties spectral pencil λI−TW
18
![Page 19: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/19.jpg)
Further analysis
Underlying concept: realization
Representation in the form
W (λ) = D + C(λIX −A)−1B (: Y → Y )
Important case: D = IY
19
![Page 20: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/20.jpg)
NB Misprint on next slide:
y(λ) = (D + C(λ−A)−1B
should be
y(λ) = (D + C(λ−A)−1B)u(λ)
20
![Page 21: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/21.jpg)
Input u Output y
21
![Page 22: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/22.jpg)
Background (2): Livsic-Brodskii characteristic function
Characteristic functions of Livsic-Brodskii type, i.e.,
IH + 2iK∗(λIG −A)−1K, KK∗ =1
2i(A−A∗)
H,G Hilbert spaces
Designed to handle operators not far from being selfadjoint
Invariant subspace problem
Echo:Bart, Gohberg, Kaashoek: Operator polynomials as inverses ofcharacteristic functions, 1978First paper in first issue of the newly founded IEOT
22
![Page 23: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/23.jpg)
Realization takes different concrete forms depending on analyt-
icity/continuity properties W (λ)
For instance:
• W (λ) analytic on bounded Cauchy domain and continuous to-
ward its boundary Γ (D identity operator)
• Wλ) analytic on bounded open set, no boundary requirement
(Mitiagin, 1978)
• Wλ) rational matrix function, analytic at infinity
(Systems Theory)
23
![Page 24: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/24.jpg)
Connection with realization
W (λ) = D + C(λIX −A)−1B
Linearization by equivalence after two-sided extension:
[W (λ) 0
0 IX
]= E(λ)
[λIX − (A−BD−1C) 0
0 IY
]F (λ)
(Many) properties W (λ)↔ properties spectral pencil λIX −A×
A×= A-BD−1C
W (λ)−1 = D−1 −D−1C(λIX −A×)−1BD−1
24
![Page 25: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/25.jpg)
Realization, factorization, invariant subspaces
W (λ) = IY + C(λIX −A)−1B
W (λ)−1 = IY -C(λIX −A×)−1B
(D = IY for simplicity)
M invariant subspace A
M× invariant subspace A× = A−BC
Matching: X = M uM×
25
![Page 26: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/26.jpg)
Induces factorization W (λ) = W1(λ)W2(λ)
W1(λ) = IY + C(λIX −A)−1(I − P )B
W2(λ) = IY + CP (λIX −A)−1B
P = projection of X = M uM× onto M× along M
W1(λ)−1 = IY − C(I − P )(λIX −A)−1B
W2(λ)−1 = IY − C(λIX −A)−1PB
Factorization Principle
Bart/Gohberg/Kaashoek and Van Dooren (1978)
26
![Page 27: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/27.jpg)
Opportunities:
• Choice realization W (λ) = D + C(λIX −A)−1B
for instance minimal
• Choice (matching) invariant subspaces M and M×
for instance spectral subspaces
Corresponds to factorizations with special properties pertinent
to the particular application at hand
• Stability of factorizations ↔ stability invariant subspaces
27
![Page 28: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/28.jpg)
Example:
The (vector-valued) Wiener-Hopf integral equation
φ(t)−∫ ∞
0k(t− s)φ(s) ds = f(t), t ≥ 0
Kernel function k ∈ Ln×n1 (−∞,∞)
Given function f ∈ Ln1[0,∞)
Desired solution function φ ∈ Ln1[0,∞)
28
![Page 29: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/29.jpg)
Associated operator H : Ln1[0,∞)→ Ln1[0,∞)
(Hφ)(t) = φ(t)−∫ ∞
0k(t− s)φ(s) ds, t ≥ 0
Symbol: W (λ) = In −∫ +∞
−∞eiλtk(t)dt
Continuous on the real line
limλ∈R, |λ|→∞ W (λ) = In (Riemann-Lebesgue)
Fredholm properties H ↔ factorization properties W (λ)
29
![Page 30: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/30.jpg)
H : Ln1[0,∞)→ Ln1[0,∞) invertible
m
W (λ) admits canonical Wiener-Hopf factorization
W (λ) = W−(λ)W+(λ)
Factors W−(λ) and W+(λ) satisfying certain analyticity,
continuity and invertibility conditions
on lower and upper half plane, respectively
Needed for effective description inverse H:
concrete knowledge W−(λ) and W+(λ)
30
![Page 31: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/31.jpg)
Application ’state space method’ involving the use of realization
Assumption: W (λ) rational n× n matrix function
Realization W (λ) = In + C(λIm −A)−1B
A no real eigenvalue (continuity on the real line)
(Real line splits the non-connected spectrum of the m × m
matrix A)
31
![Page 32: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/32.jpg)
Application Factorization Principle:
H : Ln1[0,∞)→ Ln1[0,∞) invertible
m
A× = A−BC no real eigenvalue and Cm = M uM×
• M= spectral subspace A / upper half plane
• M×= spectral subspace A× / lower half plane
32
![Page 33: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/33.jpg)
Description inverse of H:
(H−1f)(t) = f(t) +∫ ∞
0κ(t, s)f(s) ds, t ≥ 0
κ(t, s) =
+iCe−itA×PeisA×B, s < t,
–iCe−itA×(Im − P )eisA×B, s > t.
P = projection of Cm = M uM× onto M× along M
NB: semigroups entering the picture!
33
![Page 34: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/34.jpg)
Non-invertible case
Realization W (λ) = In + C(λIm −A)−1B
A no real eigenvalue
Wiener-Hopf operator H Fredholm ⇔ A× no real eigenvalue
Fredholm characteristics:
dim KerH = dim (M ∩M×)
codim ImH = codim (M +M×)
indexH = dim KerH − codim ImH = dimM − codimM×
34
![Page 35: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/35.jpg)
Situation when W (λ) = In + C(λIm − A)−1B does not allow fora canonical Wiener-Hopf factorization
Non-canonical factorization:
W (λ) = W−(λ)
(λ−iλ+i
)κ1 0 · · · 0
0(λ−iλ+i
)κ2 ...
... . . . 0
0 · · · 0(λ−iλ+i
)κn
W+(λ)
κ1 ≤ κ2 ≤ · · · ≤ κn: factorization indices (unique)
Can be described explicitly in terms of A,B,C and M,M×
35
![Page 36: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/36.jpg)
Similar approach works for
• Block Toeplitz operators
• Singular integral equations
• Riemann-Hilbert boundary value problem
36
![Page 37: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/37.jpg)
Impression of additional applications
Titles Part IV-VII from the monograph
A State Space Approach to Canonical Factorization
with Applications
(Bart, Gohberg, Kaashoek, Ran, OT 200, 2010):
• Factorization of selfadjoint matrix functions
• Riccati equations and factorization
• Factorization with symmetries
(Etc.)
37
![Page 38: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/38.jpg)
Also in the monograph: application tot the transport equation
Integro-differential equation modeling radiative transfer in stellar
atmosphere
Can be written as Wiener-Hopf integral equation with operator
valued kernel
Employs infinite dimensional version of the Factorization
Principle
Invertibility of the associated operator involves matching of two
specific spectral subspaces of two concrete self-adjoint operators
(albeit w.r.t. different inner products)
38
![Page 39: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/39.jpg)
Transport equation: instance of non-rational case
Leads up to considering situations where there is no analyticity
at infinity
Realizations W (λ) = IY +C(λIX −A)−1B involving unbounded
operators
Considerable technical difficulties
Direct sum decompositions associated with connected spectra.
39
![Page 40: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/40.jpg)
Led to new concept in semigroup theory: bisemigroup
S direct sum of of two possibly unbounded closed
operators S− and S+
−S− and +S+ generators of exponentially decaying semigroups
Bisemigroup generated by S:
E(t;S) =
−etS−, t < 0
+etS+, t > 0
Generalization of semigroup
Not to be confused with the notion of a group
40
![Page 41: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/41.jpg)
Simple Example:
A =
[i 0
0 −i
], S = −iA =
[1 0
0 −1
]
Group generated by S:
etS =
[et 0
0 e−t
], −∞ < t < +∞
Bisemigroup generated by S:
E(t;S) =
[−et 0
0 0
], −∞ < t < 0
E(t;S) =
[0 0
0 e−t
], 0 < t < +∞
41
![Page 42: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/42.jpg)
Bisemigroups introduced in BGK paper inJournal of Functional Analysis 1986
Sparked off considerable ’follow up’:
Cornelis van der Mee (student Kaashoek):Exponentially dichotomous operators and applicationsOT 182, 2008Applications: Wiener-Hopf factorization and Riccati equations,transport equations, diffusion equations of indefinite Sturm-Liouvilletype, noncausal infinite dimensional linear continuous-time sys-tems, and functional differential equations of mixed type
Semi-Plenary Talk IWOTA 2014Christian Wyss: Dichotomy, spectral subspaces and unboundedprojections
42
![Page 43: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/43.jpg)
Umbrella:
State space method in analysis
OT 200: A State Space Approach to Canonical factorization
with Applications (BGKR, 2010)
Still very much alive
Latest paper showing this in the title:
Frazho, Ter Horst, Kaashoek: State space formulas for a
suboptimal rational Leech problem I: Maximum entropy solution
(2014)
43
![Page 44: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/44.jpg)
As indicated earlier: also other ’Kaashoek topics’
Mention here:
Completion, extension and interpolation problems
General issue:
Object partly known/given
Determine missing parts such that certain conditions are satis-
fied
44
![Page 45: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/45.jpg)
Example: Positive completions of band matrices
* * * * * * * * ? ? ? ? ?
∗ * * * * * * * * ? ? ? ?
* * * * * * * * * * ? ? ?
* * * * * * * * * * * ? ?
* * * * * * * * * * * * ?
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
? * * * * * * * * * * * *
? ? * * * * * * * * * * *
? ? ? * * * * * * * * * *
? ? ? ? * * * * * * * * *
? ? ? ? ? * * * * * * * *
45
![Page 46: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/46.jpg)
Example: Strictly contractive completions
* * * * ? ? ? ?
* * * * * ? ? ?
* * * * * * ? ?
* * * * * * * ?
* * * * * * * *
(1)
Reduction to positive extension problem for band matrix:
[I3 (1)
(1)∗ I8
]
‖
46
![Page 47: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/47.jpg)
Positive extension band matrix
1 0 0 0 0 * * * * ? ? ? ?
0 1 0 0 0 * * * * * ? ? ?
0 0 1 0 0 * * * * * * ? ?
0 0 0 1 0 * * * * * * * ?
0 0 0 0 1 * * * * * * * *
* * * * * 1 0 0 0 0 0 0 0
* * * * * 0 1 0 0 0 0 0 0
* * * * * 0 0 1 0 0 0 0 0
* * * * * 0 0 0 1 0 0 0 0
? * * * * 0 0 0 0 1 0 0 0
? ? * * * 0 0 0 0 0 1 0 0
? ? ? * * 0 0 0 0 0 0 1 0
? ? ? ? * 0 0 0 0 0 0 0 1
47
![Page 48: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/48.jpg)
Band structure matrix algebra:direct sum of five linear manifolds
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * *
48
![Page 49: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/49.jpg)
Diagonal manifold (actually subalgebra)
* 0 0 0 0 0 0 0 0 0 0 0 0
0 * 0 0 0 0 0 0 0 0 0 0 0
0 0 * 0 0 0 0 0 0 0 0 0 0
0 0 0 * 0 0 0 0 0 0 0 0 0
0 0 0 0 * 0 0 0 0 0 0 0 0
0 0 0 0 0 * 0 0 0 0 0 0 0
0 0 0 0 0 0 * 0 0 0 0 0 0
0 0 0 0 0 0 0 * 0 0 0 0 0
0 0 0 0 0 0 0 0 * 0 0 0 0
0 0 0 0 0 0 0 0 0 * 0 0 0
0 0 0 0 0 0 0 0 0 0 * 0 0
0 0 0 0 0 0 0 0 0 0 0 * 0
0 0 0 0 0 0 0 0 0 0 0 0 *
49
![Page 50: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/50.jpg)
Upper band manifold
0 * * * * * * * 0 0 0 0 0
0 0 * * * * * * * 0 0 0 0
0 0 0 * * * * * * * 0 0 0
0 0 0 0 * * * * * * * 0 0
0 0 0 0 0 * * * * * * * 0
0 0 0 0 0 0 * * * * * * *
0 0 0 0 0 0 0 * * * * * *
0 0 0 0 0 0 0 0 * * * * *
0 0 0 0 0 0 0 0 0 * * * *
0 0 0 0 0 0 0 0 0 0 * * *
0 0 0 0 0 0 0 0 0 0 0 * *
0 0 0 0 0 0 0 0 0 0 0 0 *
0 0 0 0 0 0 0 0 0 0 0 0 0
50
![Page 51: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/51.jpg)
Lower band manifold = (upper band manifold)∗
0 0 0 0 0 0 0 0 0 0 0 0 0
* 0 0 0 0 0 0 0 0 0 0 0 0
* * 0 0 0 0 0 0 0 0 0 0 0
* * * 0 0 0 0 0 0 0 0 0 0
* * * * 0 0 0 0 0 0 0 0 0
* * * * * 0 0 0 0 0 0 0 0
* * * * * * 0 0 0 0 0 0 0
* * * * * * * 0 0 0 0 0 0
0 * * * * * * * 0 0 0 0 0
0 0 * * * * * * * 0 0 0 0
0 0 0 * * * * * * * 0 0 0
0 0 0 0 * * * * * * * 0 0
0 0 0 0 0 * * * * * * * 0
51
![Page 52: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/52.jpg)
Upper triangle manifold
0 0 0 0 0 0 0 0 * * * * *
0 0 0 0 0 0 0 0 0 * * * *
0 0 0 0 0 0 0 0 0 0 * * *
0 0 0 0 0 0 0 0 0 0 0 * *
0 0 0 0 0 0 0 0 0 0 0 0 *
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
52
![Page 53: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/53.jpg)
Lower triangle manifold = (upper triangle manifold)∗
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
* 0 0 0 0 0 0 0 0 0 0 0 0
* * 0 0 0 0 0 0 0 0 0 0 0
* * * 0 0 0 0 0 0 0 0 0 0
* * * * 0 0 0 0 0 0 0 0 0
* * * * * 0 0 0 0 0 0 0 0
53
![Page 54: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/54.jpg)
Can be considered for more general algebras with involution
and unit element
(via abstract general scheme)
Applications:
• Scalar matrix completion (positive / strictly contractive)
• Operator matrix completion (positive / strictly contractive)
• Caratheodory-Toeplitz extension problem
• Nevanlina-Pick interpolation
• Nehari extension problem
54
![Page 55: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/55.jpg)
Major contributions Kaashoek et al
Briefly discuss here
• rational contractive interpolants
Related to ’Nehari’
Involves State Space method (again)
Rationality requirement: important for concrete applications
(system / control theory)
55
![Page 56: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/56.jpg)
M rational m×m matrix function
Assumptions:
• M has its poles in the open unit disc D
• M analytic at ∞ and vanishes there
Implies existence stable realization
M(λ) = C(λIn −A)−1B, σ(A) ⊂ D
56
![Page 57: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/57.jpg)
Strictly contractive rational interpolant F :
• F rational without poles on T
• ‖F (ζ)‖ < 1 for all ζ in the unit circle T
• F −M analytic on the open unit disc D
57
![Page 58: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/58.jpg)
Given stable realization
M(λ) = C(λIn −A)−1B, σ(A) ⊂ D
Introduce:
Controllability Gramian: Gc =∞∑j=0
AjBB∗(A∗)j
Observability Gramian: Go =∞∑j=0
(A∗)jC∗CAj
Well-defined because σ(A) ⊂ D (stability)
58
![Page 59: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/59.jpg)
M has a strictly contractive interpolant
m
σ(GcGo) ⊂ D
Description of all strictly contractive (rational) in terms
of A,B and C
Identification of a unique one that maximizes an entropy type
integral:
the maximum entropy interpolant of M
59
![Page 60: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/60.jpg)
Marinus A. Kaashoek:
central figure in Operator Theory
Earlier slide with MathScinet List co-authors Printscreen
Count: fifty
60
![Page 61: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/61.jpg)
Services/international (selection):
• Several editorships
• Several co-editorships special issues/volumes
• Co-organizer several conferences
• Member/chairman Steering Committees MTNS and IWOTA
Will step down by the end of the year
– after becoming eighty in November!
61
![Page 62: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/62.jpg)
Services/national (selection):
• Chairman Board Dutch Mathematical Society
• Dean Faculty of Mathematics and Computer Science
VU Amsterdam
• Dean Faculty of Mathematics Sciences VU Amsterdam
• Netherlands Coordinator European Research Network Analysis
and Operators
• Member/chairman important advisory committees
Dutch mathematics
62
![Page 63: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/63.jpg)
Seminar Operator Theory / Analysis VU Amsterdam
Started 1976 . . . approximately 25 years
Every Thursday morning
Students, colleagues
Virtually all leading figures Operator Theory
Enormous stimulus
Number of PhD’s: 17
63
![Page 64: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/64.jpg)
Not to forget:
brought Israel Gohberg to Amsterdam on a systematic basis
64
![Page 65: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/65.jpg)
To quote Gerard Reve (Dutch writer, 1923-2006),
the closing sentence of his famous book The Evenings:
(Dutch original: De Avonden)
”It has been seen, it has not gone unnoticed.”
(Dutch: Het is gezien het is niet onopgemerkt gebleven.)
Honors!
65
![Page 66: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/66.jpg)
Honors International:
• Toeplitz Lecturer, Tel-Aviv, 1991
• Member of the Honorary Editorial Board of the journal Integral
Equations and Operator Theory, 2008
66
![Page 67: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/67.jpg)
Doctor Honoris Causa North-West University (Potchefstroom) South Africa, 2014
67
![Page 68: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/68.jpg)
Honors National:
• Honorary member Royal Dutch Mathematical Society, 2016
• Royal decoration: Order of the Dutch Lion
68
![Page 69: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/69.jpg)
Knight in the Royal Order of the Dutch Lion November 2002
69
![Page 70: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/70.jpg)
Rien:
Thanks for what you did for
mathematics
70
![Page 71: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/71.jpg)
Thanks for what you did for
IWOTA
71
![Page 72: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/72.jpg)
And, from the personal side:
Thanks for having been
my teacher,
and for becoming
my friend!
72
![Page 73: MARINUS A. KAASHOEK: half a century of operator theory in ... · Exponentially dichotomous operators and applications OT 182, 2008 Applications: Wiener-Hopf factorization and Riccati](https://reader036.vdocument.in/reader036/viewer/2022070810/5f0881bf7e708231d42259d3/html5/thumbnails/73.jpg)
Thank you
for
your attention!
73