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The Invariant Subspace Problem:General Operator Theory
vs.Concrete Operator Theory?
Problems and Recent Methods in Operator Theoryin Memory of Prof. James Jamison
Memphis, October 2015
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Introduction
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Introduction
r Invariant Subspace Problem
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Introduction
r Invariant Subspace Problem
T : H → H, linear and bounded
T (M) ⊂ (M), closed subspace
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Introduction
r Invariant Subspace Problem
T : H → H, linear and bounded
T (M) ⊂ (M), closed subspace
=⇒ M = {0} or M = H?
1 Finite dimensional complex Hilbert spaces.
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Introduction
r Invariant Subspace Problem
T : H → H, linear and bounded
T (M) ⊂ (M), closed subspace
=⇒ M = {0} or M = H?
r Remarks
1 Finite dimensional complex Hilbert spaces.
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Introduction
r Invariant Subspace Problem
T : H → H, linear and bounded
T (M) ⊂ (M), closed subspace
=⇒ M = {0} or M = H?
r Remarks
1 Finite dimensional complex Hilbert spaces.
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Example: R2
T =
(0 −11 0
),
respect to the canonical bases {e1, e2}.
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Example: R2
T =
(0 −11 0
),
respect to the canonical bases {e1, e2}.
T has no non-trivial invariant subspaces in R2
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Introduction
r Invariant Subspace Problem
T : H → H, linear and bounded
T (M) ⊂ (M), closed subspace
=⇒ M = {0} or M = H?
r Remarks
1 Finite dimensional complex Hilbert spaces.
2 Non-separable Hilbert spaces.
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Introduction
r Invariant Subspace Problem
T : H → H, linear and bounded
T (M) ⊂ (M), closed subspace
=⇒ M = {0} or M = H?
r Remarks
1 Finite dimensional complex Hilbert spaces.
2 Non-separable Hilbert spaces.
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Introduction
r `2 = {{an}n≥1 ⊂ C :∞∑n=1
|an|2 <∞}
{en}n≥1 canonical bases in `2
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Introduction
r `2 = {{an}n≥1 ⊂ C :∞∑n=1
|an|2 <∞}
{en}n≥1 canonical bases in `2
Sen = en+1 n ≥ 1
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Introduction
r `2 = {{an}n≥1 ⊂ C :∞∑n=1
|an|2 <∞}
{en}n≥1 canonical bases in `2
Sen = en+1 n ≥ 1
Characterization of the invariant subspaces of S?
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Introduction
r `2 = {{an}n≥1 ⊂ C :∞∑n=1
|an|2 <∞}
{en}n≥1 canonical bases in `2
Sen = en+1 n ≥ 1
Characterization of the invariant subspaces of S?
ker(S − λI ) = {0} for any λ ∈ C.
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Introduction
r `2 = {{an}n≥1 ⊂ C :∞∑n=1
|an|2 <∞}
{en}n≥1 canonical bases in `2
Sen = en+1 n ≥ 1
Characterization of the invariant subspaces of S?
ker(S − λI ) = {0} for any λ ∈ C. That is, σp(S) = ∅.
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Introduction
r `2 = {{an}n≥1 ⊂ C :∞∑n=1
|an|2 <∞}
{en}n≥1 canonical bases in `2
Sen = en+1 n ≥ 1
Characterization of the invariant subspaces of S?
Classical Beurling Theory:Inner-Outer Factorization of the functions in the Hardy space.
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Introduction
r Classes of operators with known invariant subspaces:
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Introduction
r Classes of operators with known invariant subspaces:
? Normal operators (Spectral Theorem)
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Introduction
r Classes of operators with known invariant subspaces:
? Normal operators (Spectral Theorem)
? Compact Operators
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Introduction
r Classes of operators with known invariant subspaces:
? Normal operators (Spectral Theorem)
? Compact Operators
σ(T ) = {λj}j≥1 ∪ {0}
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Introduction
r Classes of operators with known invariant subspaces:
? Normal operators (Spectral Theorem)
? Compact Operators
σ(T ) = {λj}j≥1 ∪ {0}
∗ 1951, J. von Newman, (Hilbert space case)
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Introduction
r Classes of operators with known invariant subspaces:
? Normal operators (Spectral Theorem)
? Compact Operators
σ(T ) = {λj}j≥1 ∪ {0}
∗ 1951, J. von Newman, (Hilbert space case)
∗ 1954, Aronszajn and Smith (general case)
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Introduction
r Classes of operators with known invariant subspaces:
? Normal operators (Spectral Theorem)
? Compact Operators
? Polynomially Compact Operators
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Introduction
r Classes of operators with known invariant subspaces:
? Normal operators (Spectral Theorem)
? Compact Operators
? Polynomially Compact Operators
∗ 1966, Bernstein and Robinson, (Hilbert space case)
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Introduction
r Classes of operators with known invariant subspaces:
? Normal operators (Spectral Theorem)
? Compact Operators
? Polynomially Compact Operators
∗ 1966, Bernstein and Robinson, (Hilbert space case)
∗ 1967, Halmos
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Introduction
r Classes of operators with known invariant subspaces:
? Normal operators (Spectral Theorem)
? Compact Operators
? Polynomially Compact Operators
∗ 1966, Bernstein and Robinson, (Hilbert space case)
∗ 1967, Halmos
∗ 1960’s, Gillespie, Hsu, Kitano...
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In the Banach space setting
In 1975, P. Enflo showed in the Seminaire Maurey-Schwartz at theEcole Polytechnique in Parıs:
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In the Banach space setting
In 1975, P. Enflo showed in the Seminaire Maurey-Schwartz at theEcole Polytechnique in Parıs:
There exists a separable Banach space B and a linear, boundedoperator T acting on B, inyective and with dense range,
without no non-trivial closed invariant subspaces.
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In the Banach space setting
In 1975, P. Enflo showed in the Seminaire Maurey-Schwartz at theEcole Polytechnique in Parıs:
There exists a separable Banach space B and a linear, boundedoperator T acting on B, inyective and with dense range,
without no non-trivial closed invariant subspaces.
1987, P. Enflo “On the invariant subspace problem for Banachspaces”, Acta Math. 158 (1987), no. 3-4, 213-313.
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In the Banach space setting
In 1975, P. Enflo showed in the Seminaire Maurey-Schwartz at theEcole Polytechnique in Parıs:
There exists a separable Banach space B and a linear, boundedoperator T acting on B, inyective and with dense range,
without no non-trivial closed invariant subspaces.
1987, P. Enflo “On the invariant subspace problem for Banachspaces”, Acta Math. 158 (1987), no. 3-4, 213-313.
r 1985, C. Read, Construction of a linear bounded operator on `1
without non-trivial closed invariant subspaces.
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The big open question
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The big open question
Does every linear bounded operator T acting on a separable,reflexive complex Banach space B (or a Hilbert space H) have a
non-trivial closed invariant subspace?
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Classes of operators with known invariant subspaces
r 1973, Lomonosov
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Classes of operators with known invariant subspaces
r 1973, Lomonosov
Theorem (Lomonosov) Let T be a linear bounded operator onH, T 6= CId . If T commutes with a non-null compact operator,then T has a non-trivial closed invariant subspace.
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Classes of operators with known invariant subspaces
r 1973, Lomonosov
Theorem (Lomonosov) Let T be a linear bounded operator onH, T 6= CId . If T commutes with a non-null compact operator,then T has a non-trivial closed invariant subspace. Moreover, Thas a non-trivial closed hyperinvariant subspace.
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Classes of operators with known invariant subspaces
r 1973, Lomonosov
Theorem (Lomonosov) Let T be a linear bounded operator onH, T 6= CId . If T commutes with a non-null compact operator,then T has a non-trivial closed invariant subspace. Moreover, Thas a non-trivial closed hyperinvariant subspace.
Theorem (Lomonosov) Any linear bounded operator T , not amultiple of the identity, has a nontrivial invariant closed subspace ifit commutes with a non-scalar operator that commutes with anonzero compact operator.
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Classes of operators with known invariant subspaces
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Classes of operators with known invariant subspaces
r Does every operator satisfy “Lomonosov Hypotheses”?
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Classes of operators with known invariant subspaces
r Does every operator satisfy “Lomonosov Hypotheses”?
r 1980, Hadwin; Nordgren; Radjavi y Rosenthal
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Classes of operators with known invariant subspaces
r Does every operator satisfy “Lomonosov Hypotheses”?
r 1980, Hadwin; Nordgren; Radjavi y Rosenthal
Construction of a ”quasi-analytic” shift S on a weighted `2 spacewhich has the following property: if K is a compact operator whichcommutes with a nonzero, non scalar operator in the commutantof S , then K = 0.
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A “Concrete Operator Theory” approach
r Universal Operators (in the sense of G. C. Rota)
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A “Concrete Operator Theory” approach
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M ,
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A “Concrete Operator Theory” approach
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M , i. e., λT = J−1UJwhere J : H →M is a linear isomorphism.
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A “Concrete Operator Theory” approach
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M , i. e., λT = J−1UJwhere J : H →M is a linear isomorphism.
r Example. Adjoint of a unilateral shift of infinite multiplicity.
![Page 46: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/46.jpg)
A “Concrete Operator Theory” approach
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M , i. e., λT = J−1UJwhere J : H →M is a linear isomorphism.
r Example. Adjoint of a unilateral shift of infinite multiplicity. Itmay be regarded as S∗ in (`2(H)) defined by
S∗((h0,h1, h2, · · · )) = (h1,h2, · · · )
for (h0, h1,h2, · · · ) ∈ `2(H).
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A “Concrete Operator Theory” approach
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M , i. e., λT = J−1UJwhere J : H →M is a linear isomorphism.
r Example. Adjoint of a unilateral shift of infinite multiplicity. Itmay be regarded as S∗ in (`2(H)) defined by
S∗((h0,h1, h2, · · · )) = (h1,h2, · · · )
for (h0, h1,h2, · · · ) ∈ `2(H).
r Example. Let a > 0 and Ta : L2(0,∞)→ L2(0,∞) defined by
Taf(t) = f(t + a), for t > 0.
Ta is universal.
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A “Concrete Operator Theory” approach: universal operators
r Proposition. Let H be a Hilbert space and U a linear boundedoperator. Suppose that U is a universal operator. The followingconditions are equivalent:
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A “Concrete Operator Theory” approach: universal operators
r Proposition. Let H be a Hilbert space and U a linear boundedoperator. Suppose that U is a universal operator. The followingconditions are equivalent:
1. Every linear bounded operator T on H has a non-trivial closedinvariant subspace.
![Page 50: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/50.jpg)
A “Concrete Operator Theory” approach: universal operators
r Proposition. Let H be a Hilbert space and U a linear boundedoperator. Suppose that U is a universal operator. The followingconditions are equivalent:
1. Every linear bounded operator T on H has a non-trivial closedinvariant subspace.
2. Every closed invariant subspace M of U of dimension greaterthan 1 contains a proper closed and invariant subspace
![Page 51: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/51.jpg)
A “Concrete Operator Theory” approach: universal operators
r Proposition. Let H be a Hilbert space and U a linear boundedoperator. Suppose that U is a universal operator. The followingconditions are equivalent:
1. Every linear bounded operator T on H has a non-trivial closedinvariant subspace.
2. Every closed invariant subspace M of U of dimension greaterthan 1 contains a proper closed and invariant subspace (i.e. theminimal non-trivial closed and invariant subspaces for U areone-dimensional).
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Providing universal operators
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M , i. e., λT = J−1UJwhere J : H →M is a linear isomorphism.
1 Ker(U) is infinite dimensional,
2 U is surjective.
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Providing universal operators
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M , i. e., λT = J−1UJwhere J : H →M is a linear isomorphism.
r 1969, Caradus
1 Ker(U) is infinite dimensional,
2 U is surjective.
![Page 54: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/54.jpg)
Providing universal operators
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M , i. e., λT = J−1UJwhere J : H →M is a linear isomorphism.
r 1969, Caradus
Let U be a linear bounded operator on a Hilbert space. Assumethat
1 Ker(U) is infinite dimensional,
2 U is surjective.
![Page 55: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/55.jpg)
Providing universal operators
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M , i. e., λT = J−1UJwhere J : H →M is a linear isomorphism.
r 1969, Caradus
Let U be a linear bounded operator on a Hilbert space. Assumethat
1 Ker(U) is infinite dimensional,
2 U is surjective.
![Page 56: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/56.jpg)
Providing universal operators
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M , i. e., λT = J−1UJwhere J : H →M is a linear isomorphism.
r 1969, Caradus
Let U be a linear bounded operator on a Hilbert space. Assumethat
1 Ker(U) is infinite dimensional,
2 U is surjective.
![Page 57: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/57.jpg)
Providing universal operators
r Universal Operators (in the sense of G. C. Rota)
A linear bounded operator U in a Hilbert space H is universal iffor any linear bounded operator T in H, there exists λ ∈ C andM∈ Lat(U) such that λT is similar to U |M , i. e., λT = J−1UJwhere J : H →M is a linear isomorphism.
r 1969, Caradus
Let U be a linear bounded operator on a Hilbert space. Assumethat
1 Ker(U) is infinite dimensional,
2 U is surjective.
Then U is universal.
![Page 58: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/58.jpg)
Idea of the Proof
Write K =Ker U
1 UV = Id,
2 UW = 0,
3 kerW = {0},4 ImW = K and ImV = K⊥.
• M =Im J is a closed subspace of U.
• J is an isomorphism onto M.
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Idea of the Proof
Write K =Ker U
Step 1: Construct V,W ∈ L(H) such that
1 UV = Id,
2 UW = 0,
3 kerW = {0},4 ImW = K and ImV = K⊥.
• M =Im J is a closed subspace of U.
• J is an isomorphism onto M.
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Idea of the Proof
Write K =Ker U
Step 1: Construct V,W ∈ L(H) such that
1 UV = Id,
2 UW = 0,
3 kerW = {0},4 ImW = K and ImV = K⊥.
• M =Im J is a closed subspace of U.
• J is an isomorphism onto M.
![Page 61: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/61.jpg)
Idea of the Proof
Write K =Ker U
Step 1: Construct V,W ∈ L(H) such that
1 UV = Id,
2 UW = 0,
3 kerW = {0},4 ImW = K and ImV = K⊥.
Step 2: Prove that U is universal.
• M =Im J is a closed subspace of U.
• J is an isomorphism onto M.
![Page 62: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/62.jpg)
Idea of the Proof
Write K =Ker U
Step 1: Construct V,W ∈ L(H) such that
1 UV = Id,
2 UW = 0,
3 kerW = {0},4 ImW = K and ImV = K⊥.
Step 2: Prove that U is universal.Let T be a linear bounded operator on H.
• M =Im J is a closed subspace of U.
• J is an isomorphism onto M.
![Page 63: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/63.jpg)
Idea of the Proof
Write K =Ker U
Step 1: Construct V,W ∈ L(H) such that
1 UV = Id,
2 UW = 0,
3 kerW = {0},4 ImW = K and ImV = K⊥.
Step 2: Prove that U is universal.Let T be a linear bounded operator on H.Let λ 6= 0 such that |λ| ‖T‖ ‖U‖ < 1 and define
J =∞∑k=0
λkVkWTk.
• M =Im J is a closed subspace of U.
• J is an isomorphism onto M.
![Page 64: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/64.jpg)
Idea of the Proof
Write K =Ker U
Step 1: Construct V,W ∈ L(H) such that
1 UV = Id,
2 UW = 0,
3 kerW = {0},4 ImW = K and ImV = K⊥.
Step 2: Prove that U is universal.Let T be a linear bounded operator on H.Let λ 6= 0 such that |λ| ‖T‖ ‖U‖ < 1 and define
J =∞∑k=0
λkVkWTk.
J satisfies J = W + λVJT and therefore, UJ = λJT.
• M =Im J is a closed subspace of U.
• J is an isomorphism onto M.
![Page 65: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/65.jpg)
Idea of the Proof
Write K =Ker U
Step 1: Construct V,W ∈ L(H) such that
1 UV = Id,
2 UW = 0,
3 kerW = {0},4 ImW = K and ImV = K⊥.
Step 2: Prove that U is universal.Let T be a linear bounded operator on H.Let λ 6= 0 such that |λ| ‖T‖ ‖U‖ < 1 and define
J =∞∑k=0
λkVkWTk.
J satisfies J = W + λVJT and therefore, UJ = λJT. In addition,
• M =Im J is a closed subspace of U.
• J is an isomorphism onto M.
![Page 66: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/66.jpg)
Examples of universal operatorsr Theorem (1987, Nordgren, Rosenthal y Wintrobe)
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Examples of universal operatorsr Theorem (1987, Nordgren, Rosenthal y Wintrobe)
Let ϕ be a hyperbolic automorphism of D. For every λ in theinterior of the spectrum of Cϕ, Cϕ − λI is universal in H2.
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Examples of universal operatorsr Theorem (1987, Nordgren, Rosenthal y Wintrobe)
Let ϕ be a hyperbolic automorphism of D. For every λ in theinterior of the spectrum of Cϕ, Cϕ − λI is universal in H2.
r Disc automorphisms
ϕ(z) = eiθp− z
1− pz(z ∈ D).
where p ∈ D and −π < θ ≤ π.
![Page 69: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/69.jpg)
Examples of universal operatorsr Theorem (1987, Nordgren, Rosenthal y Wintrobe)
Let ϕ be a hyperbolic automorphism of D. For every λ in theinterior of the spectrum of Cϕ, Cϕ − λI is universal in H2.
r Disc automorphisms
ϕ(z) = eiθp− z
1− pz(z ∈ D).
where p ∈ D and −π < θ ≤ π.
? Parabolic.
? Hyperbolic.
? Elliptic.
![Page 70: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/70.jpg)
Examples of universal operatorsr Theorem (1987, Nordgren, Rosenthal y Wintrobe)
Let ϕ be a hyperbolic automorphism of D. For every λ in theinterior of the spectrum of Cϕ, Cϕ − λI is universal in H2.
r Disc automorphisms
ϕ(z) = eiθp− z
1− pz(z ∈ D).
where p ∈ D and −π < θ ≤ π.
? Parabolic. ϕ has just one fixed point α ∈ ∂D (⇔ |p| = cos(θ/2))
? Hyperbolic.
? Elliptic.
![Page 71: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/71.jpg)
Examples of universal operatorsr Theorem (1987, Nordgren, Rosenthal y Wintrobe)
Let ϕ be a hyperbolic automorphism of D. For every λ in theinterior of the spectrum of Cϕ, Cϕ − λI is universal in H2.
r Disc automorphisms
ϕ(z) = eiθp− z
1− pz(z ∈ D).
where p ∈ D and −π < θ ≤ π.
? Parabolic. ϕ has just one fixed point α ∈ ∂D (⇔ |p| = cos(θ/2))
? Hyperbolic. ϕ has two fixed points α and β, such thatα, β ∈ ∂D (⇔ |p| > cos(θ/2))
? Elliptic.
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Examples of universal operatorsr Theorem (1987, Nordgren, Rosenthal y Wintrobe)
Let ϕ be a hyperbolic automorphism of D. For every λ in theinterior of the spectrum of Cϕ, Cϕ − λI is universal in H2.
r Disc automorphisms
ϕ(z) = eiθp− z
1− pz(z ∈ D).
where p ∈ D and −π < θ ≤ π.
? Parabolic. ϕ has just one fixed point α ∈ ∂D (⇔ |p| = cos(θ/2))
? Hyperbolic. ϕ has two fixed points α and β, such thatα, β ∈ ∂D (⇔ |p| > cos(θ/2))
? Elliptic. ϕ has two fixed points α and β, with α ∈ D(⇔ |p| < cos(θ/2))
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Examples of universal operators
r Theorem (1987, Nordgren, Rosenthal y Wintrobe)
Let ϕ be a hyperbolic automorphism of D. For every λ in theinterior of the spectrum of Cϕ, Cϕ − λI is universal in H2.
Let ϕ be a hyperbolic automorphism of D.
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Examples of universal operators
r Theorem (1987, Nordgren, Rosenthal y Wintrobe)
Let ϕ be a hyperbolic automorphism of D. For every λ in theinterior of the spectrum of Cϕ, Cϕ − λI is universal in H2.
Let ϕ be a hyperbolic automorphism of D.
We may assume that ϕ fixes 1 and −1.
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Examples of universal operators
r Theorem (1987, Nordgren, Rosenthal y Wintrobe)
Let ϕ be a hyperbolic automorphism of D. For every λ in theinterior of the spectrum of Cϕ, Cϕ − λI is universal in H2.
Let ϕ be a hyperbolic automorphism of D.
We may assume that ϕ fixes 1 and −1. Then,
ϕ(z) =z + r
1 + rz, 0 < r < 1.
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Examples of universal operators
Every linear bounded operator T has a closednon-trivial invariant subspace
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Examples of universal operators
Every linear bounded operator T has a closednon-trivial invariant subspace
m
for any f ∈ H2, not an eigenfunction of Cϕ, there existsg ∈ span{Cn
ϕf : n ≥ 0} such that g 6= 0 and
span{Cnϕg : n ≥ 0} 6= span{Cn
ϕf : n ≥ 0}.
![Page 78: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/78.jpg)
Examples of universal operators
Every linear bounded operator T has a closednon-trivial invariant subspace
m
for any f ∈ H2, not an eigenfunction of Cϕ, there existsg ∈ span{Cn
ϕf : n ≥ 0} such that g 6= 0 and
span{Cnϕg : n ≥ 0} 6= span{Cn
ϕf : n ≥ 0}.
m
the minimal non-trivial closed invariant
subspaces for Cϕ are one-dimensional
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Question
Which conditions on f ensure that Kf := span{Cnϕf : n ≥ 0} is (or
not) minimal?
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Minimal invariant subspaces
Theorem (Matache, 1993) Let ϕ be a hyperbolic automorphism ofD and 1 one of the fixed points of ϕ. Let f ∈ H2 a non-constantfunction such that f extends continuously to 1 and f(1) 6= 0. ThenKf is not minimal.
Theorem (Mortini, 1995) Let ϕ be a hyperbolic automorphism ofD and 1 the attractive fixed point of ϕ. Let f ∈ H∞ anon-constant function such that there exists the radial limit at 1and f(1) 6= 0. Then Kf is not minimal.
Theorem (Matache, 1998) Let ϕ be a hyperbolic automorphism ofD and 1 one of the fixed points of ϕ. Let f ∈ H2 a non-constantfunction such that there exists the radial limit of f at 1 andf(1) 6= 0. Assume that f is essentially bounded in an arc γ ⊂ ∂D sothat 1 ∈ γ. Then Kf is not minimal.
![Page 81: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/81.jpg)
Minimal invariant subspaces
1 lımz→−1
|f(z)| <∞,
2 |f(z)| ≤ C|z− 1|ε for some constant C, and ε > 0 in aneighborhood of 1.
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Minimal invariant subspaces
Theorem (Chkliar, 1997) Let ϕ be a hyperbolic automorphism ofD with fixed points 1 and −1. Assume 1 is the attractive fixedpoint of ϕ. Suppose that f ∈ H2 such that
1 lımz→−1
|f(z)| <∞,
2 |f(z)| ≤ C|z− 1|ε for some constant C, and ε > 0 in aneighborhood of 1.
Then the point spectrum of Cϕ acting on span{Cnϕf : n ∈ Z} H
2
contains the annulus
{z ∈ C : |ϕ′(1)|mın{ε, 12} < |z| < 1},
except, possibly, a discrete subset.
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Minimal invariant subspaces
r Theorem (2011, Shapiro) Let ϕ be a hyperbolic automorphismof D with fixed points 1 and −1.
1 If f ∈√
(z− 1)(z + 1)H2 \ {0}, then the point spectrum of
Cϕ acting on span{Cnϕf : n ∈ Z} H
2
intersects the unit circlein a set of positive measure.
2 If f ∈√
(z− 1)(z + 1)Hp \ {0} for some p > 2, then the
point spectrum of Cϕ acting on span{Cnϕf : n ∈ Z} H
2
contains an open annulus centered at the origin.
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Minimal invariant subspaces
Theorem. (2011, GG-Gorkin) Let ϕ be a hyperbolic automorphismof D fixing 1 and −1. Assume that 1 is the attractive fixed point.Let f be a nonzero function in H2 that is continuous at 1 and −1and such that f(1) = f(−1) = 0. Then there exists g ∈ H2
satisfying the following conditions:
1 g ∈ Kf := span{Cnϕf : n ≥ 0} H
2
.
2 There exists a subsequence {ϕnk} such that g ◦ ϕ−nkconverges to f uniformly on compact subsets of D as k→∞.
3 g has no radial limit at −1.
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Minimal invariant subspaces
Theorem. (2011, GG-Gorkin) Let ϕ be a hyperbolic automorphismof D fixing 1 and −1. Assume that 1 is the attractive fixed point.Let f be a nonzero function in H2 that is continuous at 1 and −1and such that f(1) = f(−1) = 0. Then there exists g ∈ H2
satisfying the following conditions:
1 g ∈ Kf := span{Cnϕf : n ≥ 0} H
2
.
2 There exists a subsequence {ϕnk} such that g ◦ ϕ−nkconverges to f uniformly on compact subsets of D as k→∞.
3 g has no radial limit at −1.
Furthermore, if f belongs to the disc algebra A(D), then g ∈ H∞and, consequently, if Kf is minimal, then Kg = Kf .
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Question
Eigenfuntions of Cϕ?
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Question
Eigenfuntions of Cϕ?
r 2012, GG, Gorkin and Suarez, Constructive characterization ofeigenfunctions of Cϕ in the Hardy spaces Hp
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Universal operators vs. Lomonosov Theorem
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Universal operators vs. Lomonosov Theorem
r Naive Question: Does there exist a universal operator whichconmutes with a non-null compact operator in a non-trivial way?
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Universal operatorsSuppose S is a multiplication operator on the Hardy space H2
whose symbol is a singular inner function or infinite Blaschkeproduct.
1 S is an isometric operator.
2 S∗ has infinite dimensional kernel and maps H2 onto H2.
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Universal operatorsSuppose S is a multiplication operator on the Hardy space H2
whose symbol is a singular inner function or infinite Blaschkeproduct.r Remarks
1 S is an isometric operator.
2 S∗ has infinite dimensional kernel and maps H2 onto H2.
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Universal operatorsSuppose S is a multiplication operator on the Hardy space H2
whose symbol is a singular inner function or infinite Blaschkeproduct.r Remarks
1 S is an isometric operator.
2 S∗ has infinite dimensional kernel and maps H2 onto H2.
![Page 93: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/93.jpg)
Universal operatorsSuppose S is a multiplication operator on the Hardy space H2
whose symbol is a singular inner function or infinite Blaschkeproduct.r Remarks
1 S is an isometric operator.
2 S∗ has infinite dimensional kernel and maps H2 onto H2.
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Universal operatorsSuppose S is a multiplication operator on the Hardy space H2
whose symbol is a singular inner function or infinite Blaschkeproduct.r Remarks
• S∗ is universal.
• Using the Wold Decomposition Theorem, such an operatorcan be represented as a block matrix on H = ⊕∞k=1S
kW,where W = H2 SH2. Such a matrix is an upper triangularand has the identity on the super-diagonal:
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Universal operatorsSuppose S is a multiplication operator on the Hardy space H2
whose symbol is a singular inner function or infinite Blaschkeproduct.r Remarks
• S∗ is universal.
• Using the Wold Decomposition Theorem, such an operatorcan be represented as a block matrix on H = ⊕∞k=1S
kW,where W = H2 SH2. Such a matrix is an upper triangularand has the identity on the super-diagonal:
![Page 96: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/96.jpg)
Universal operatorsSuppose S is a multiplication operator on the Hardy space H2
whose symbol is a singular inner function or infinite Blaschkeproduct.r Remarks
• S∗ is universal.
• Using the Wold Decomposition Theorem, such an operatorcan be represented as a block matrix on H = ⊕∞k=1S
kW,where W = H2 SH2. Such a matrix is an upper triangularand has the identity on the super-diagonal:
S∗ ∼
0 I 0 0 · · ·0 0 I 0 · · ·0 0 0 I · · ·
. . .
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Universal operatorsAn easy computation shows that every operator that commuteswith S∗ has the form
• This is an upper triangular block matrix whose entries on eachdiagonal are the same operator on the infinite dimensionalHilbert space W.
• Every block in such a matrix occurs infinitely often.
• So, the only compact operator that commutes with theuniversal operator S∗ is 0,
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Universal operatorsAn easy computation shows that every operator that commuteswith S∗ has the form
A ∼
A0 A−1 A−2 A−3 · · ·0 A0 A−1 A−2 · · ·0 0 A0 A−1 · · ·
. . .
an upper triangular block Toeplitz matrix.
Observe that:
• This is an upper triangular block matrix whose entries on eachdiagonal are the same operator on the infinite dimensionalHilbert space W.
• Every block in such a matrix occurs infinitely often.
• So, the only compact operator that commutes with theuniversal operator S∗ is 0,
![Page 99: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/99.jpg)
Universal operatorsAn easy computation shows that every operator that commuteswith S∗ has the form
A ∼
A0 A−1 A−2 A−3 · · ·0 A0 A−1 A−2 · · ·0 0 A0 A−1 · · ·
. . .
an upper triangular block Toeplitz matrix.
Observe that:
• This is an upper triangular block matrix whose entries on eachdiagonal are the same operator on the infinite dimensionalHilbert space W.
• Every block in such a matrix occurs infinitely often.
• So, the only compact operator that commutes with theuniversal operator S∗ is 0,
![Page 100: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/100.jpg)
Universal operatorsAn easy computation shows that every operator that commuteswith S∗ has the form
A ∼
A0 A−1 A−2 A−3 · · ·0 A0 A−1 A−2 · · ·0 0 A0 A−1 · · ·
. . .
an upper triangular block Toeplitz matrix.
Observe that:
• This is an upper triangular block matrix whose entries on eachdiagonal are the same operator on the infinite dimensionalHilbert space W.
• Every block in such a matrix occurs infinitely often.
• So, the only compact operator that commutes with theuniversal operator S∗ is 0,
![Page 101: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/101.jpg)
Universal operatorsAn easy computation shows that every operator that commuteswith S∗ has the form
A ∼
A0 A−1 A−2 A−3 · · ·0 A0 A−1 A−2 · · ·0 0 A0 A−1 · · ·
. . .
an upper triangular block Toeplitz matrix.
Observe that:
• This is an upper triangular block matrix whose entries on eachdiagonal are the same operator on the infinite dimensionalHilbert space W.
• Every block in such a matrix occurs infinitely often.
• So, the only compact operator that commutes with theuniversal operator S∗ is 0,
![Page 102: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/102.jpg)
Universal operatorsAn easy computation shows that every operator that commuteswith S∗ has the form
A ∼
A0 A−1 A−2 A−3 · · ·0 A0 A−1 A−2 · · ·0 0 A0 A−1 · · ·
. . .
an upper triangular block Toeplitz matrix.
Observe that:
• This is an upper triangular block matrix whose entries on eachdiagonal are the same operator on the infinite dimensionalHilbert space W.
• Every block in such a matrix occurs infinitely often.
• So, the only compact operator that commutes with theuniversal operator S∗ is 0, not an interesting compactoperator!
![Page 103: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/103.jpg)
Universal operatorsr Theorem (2011, Cowen-GG)
Let ϕ be a hyperbolic automorphism of D. Then C∗ϕ is similar tothe Toeplitz operator Tψ, where ψ is the covering map of the unitdisc onto the interior of the spectrum σ(Cϕ).
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Universal operatorsr Theorem (2011, Cowen-GG)
Let ϕ be a hyperbolic automorphism of D. Then C∗ϕ is similar tothe Toeplitz operator Tψ, where ψ is the covering map of the unitdisc onto the interior of the spectrum σ(Cϕ).
r Theorem (1980, Cowen)
A Toeplitz operator Tψ in H2, where ψ ∈ H∞ is an inner functionor a covering map conmutes with a compact operator K if andonly if K = 0.
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Universal operatorsr Theorem (2011, Cowen-GG)
Let ϕ be a hyperbolic automorphism of D. Then C∗ϕ is similar tothe Toeplitz operator Tψ, where ψ is the covering map of the unitdisc onto the interior of the spectrum σ(Cϕ).
r Theorem (1980, Cowen)
A Toeplitz operator Tψ in H2, where ψ ∈ H∞ is an inner functionor a covering map conmutes with a compact operator K if andonly if K = 0.r Straightforward consequence
Known universal operators are not commuting with non-nullcompact operators.
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Universal operators
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Universal operators
r Theorem [2013, Cowen-GG] There exists a universal operatorwhich commutes with an injective, dense range compact operator.
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A universal operator which commutes with a compact operator
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Compact operators commuting with universal operators
We have seen that some universal operators commute with acompact operator and others do not.
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Compact operators commuting with universal operators
We have seen that some universal operators commute with acompact operator and others do not.
Observation: There are many more compact operators than justone commuting with the universal operator T∗ϕ.
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Compact operators commuting with universal operators
We have seen that some universal operators commute with acompact operator and others do not.
Observation: There are many more compact operators than justone commuting with the universal operator T∗ϕ.
Definition. Let Kϕ be the set of compact operators that commutewith T∗ϕ, that is,
Kϕ = {G ∈ B(H2) : G is compact, and T∗ϕG = GT∗ϕ}
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Compact operators commuting with universal operators
We have seen that some universal operators commute with acompact operator and others do not.
Observation: There are many more compact operators than justone commuting with the universal operator T∗ϕ.
Definition. Let Kϕ be the set of compact operators that commutewith T∗ϕ, that is,
Kϕ = {G ∈ B(H2) : G is compact, and T∗ϕG = GT∗ϕ}
Remark. Kϕ 6= (0).
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Compact operators commuting with universal operators
If F is a bounded operator on H2, we will write {F}′ for thecommutant of F, the set of operators that commute with F, thatis,
{F}′ = {G ∈ B(H2) : GF = FG}.
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Compact operators commuting with universal operators
If F is a bounded operator on H2, we will write {F}′ for thecommutant of F, the set of operators that commute with F, thatis,
{F}′ = {G ∈ B(H2) : GF = FG}.
For any operator F, the commutant {F}′ is a norm-closedsubalgebra of B(H2).
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Compact operators commuting with universal operators
Theorem [2015, Cowen, GG] The set Kϕ is a closed subalgebraof {T∗ϕ}′ that is a two-sided ideal in {T∗ϕ}′. In particular, if G is acompact operator in Kϕ and g and h are bounded analyticfunctions on the disk, then T∗gG, GT∗h , and T∗gGT∗h are all in Kϕ.Moreover, every operator in Kϕ is quasi-nilpotent.
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Consequences, Further Observations, and a Question
Let A be a linear bounded operator on a Hilbert space and T auniversal operator which commutes with a compact operator W.
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Consequences, Further Observations, and a Question
Let A be a linear bounded operator on a Hilbert space and T auniversal operator which commutes with a compact operator W.
• WLOG A IS the restriction of T to M.
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Consequences, Further Observations, and a Question
Let A be a linear bounded operator on a Hilbert space and T auniversal operator which commutes with a compact operator W.
• WLOG A IS the restriction of T to M.
• WLOG M 6= H because if so, Lomonosov gives hyperinvariantsubspace
![Page 119: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/119.jpg)
Consequences, Further Observations, and a Question
Let A be a linear bounded operator on a Hilbert space and T auniversal operator which commutes with a compact operator W.
• WLOG A IS the restriction of T to M.
• WLOG M 6= H because if so, Lomonosov gives hyperinvariantsubspace
• WLOG T and A are invertible: replace T by T + (1 + ‖T‖)I
![Page 120: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/120.jpg)
Consequences, Further Observations, and a Question
Let A be a linear bounded operator on a Hilbert space and T auniversal operator which commutes with a compact operator W.
• WLOG A IS the restriction of T to M.
• WLOG M 6= H because if so, Lomonosov gives hyperinvariantsubspace
• WLOG T and A are invertible: replace T by T + (1 + ‖T‖)I
• H = M⊕M⊥ and with respect to this decomposition
T ∼(
A B0 C
)and W ∼
(P QR S
)where A, C are invertible and P, Q, R, S are compact.
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Consequences, Further Observations, and a Question
Let A be a linear bounded operator on a Hilbert space and T auniversal operator which commutes with a compact operator W.
• WLOG A IS the restriction of T to M.
• WLOG M 6= H because if so, Lomonosov gives hyperinvariantsubspace
• WLOG T and A are invertible: replace T by T + (1 + ‖T‖)I
• H = M⊕M⊥ and with respect to this decomposition
T ∼(
A B0 C
)and W ∼
(P QR S
)where A, C are invertible and P, Q, R, S are compact.
• NOT P = 0 and R = 0 because kernel(W) = (0).
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Consequences, Further Observations, and a Question
• From the relation, TW = WT, it follows that
AP + BR = PA and CR = RA
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Consequences, Further Observations, and a Question
• From the relation, TW = WT, it follows that
AP + BR = PA and CR = RA
Observation: Since A is the operator of primary interest, Equation
AP + BR = PA
is not so interesting if P = 0.
![Page 124: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/124.jpg)
Consequences, Further Observations, and a Question
r Lemma. If the universal operator T = T∗ϕ and the compactoperator W = W∗ψ,J have the representations
T ∼(
A B0 C
)and W ∼
(P QR S
)respect H = M⊕M⊥, then there are a universal operator T andan injective compact operator W with dense range that commutefor which P in a replacement of P is not zero, that is, without lossof generality, we may assume P 6= 0.
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Consequences, Further Observations, and a Question
r Theorem [Cowen, GG] Let the universal operator T and thecommuting injective compact operator W with dense range havingthe representations with P 6= 0. Then the following are true:
• Either R 6= 0 or A has a nontrivial hyperinvariant subspace.
• Either ker(R) = (0) or A has a nontrivial invariant subspace.
• Either B 6= 0 or A has a nontrivial hyperinvariant subspace.
![Page 126: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/126.jpg)
Consequences, Further Observations, and a Question
r Theorem [Cowen, GG] Let the universal operator T and thecommuting injective compact operator W with dense range havingthe representations with P 6= 0. Then the following are true:
• Either R 6= 0 or A has a nontrivial hyperinvariant subspace.
• Either ker(R) = (0) or A has a nontrivial invariant subspace.
• Either B 6= 0 or A has a nontrivial hyperinvariant subspace.
![Page 127: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/127.jpg)
Consequences, Further Observations, and a Question
r Theorem [Cowen, GG] Let the universal operator T and thecommuting injective compact operator W with dense range havingthe representations with P 6= 0. Then the following are true:
• Either R 6= 0 or A has a nontrivial hyperinvariant subspace.
• Either ker(R) = (0) or A has a nontrivial invariant subspace.
• Either B 6= 0 or A has a nontrivial hyperinvariant subspace.
![Page 128: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/128.jpg)
Consequences, Further Observations, and a Question
r Theorem [Cowen, GG] Let the universal operator T and thecommuting injective compact operator W with dense range havingthe representations with P 6= 0. Then the following are true:
• Either R 6= 0 or A has a nontrivial hyperinvariant subspace.
• Either ker(R) = (0) or A has a nontrivial invariant subspace.
• Either B 6= 0 or A has a nontrivial hyperinvariant subspace.
![Page 129: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/129.jpg)
Consequences, Further Observations, and a Question
Theorem [Cowen, GG] Suppose L is an invariant subspace forthe universal operator T∗ϕ and the block matrix(
A B0 C
)represents T∗ϕ based on the splitting H2 = M⊕M⊥. Then, the
projection of L⊥ onto M is an invariant linear manifold for A∗, theadjoint of the restriction of T∗ϕ to M.
![Page 130: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/130.jpg)
Consequences, Further Observations, and a Question
Theorem [Cowen, GG] Suppose L is an invariant subspace forthe universal operator T∗ϕ and the block matrix(
A B0 C
)represents T∗ϕ based on the splitting H2 = M⊕M⊥. Then, the
projection of L⊥ onto M is an invariant linear manifold for A∗, theadjoint of the restriction of T∗ϕ to M.
![Page 131: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/131.jpg)
Consequences, Further Observations, and a Question
Theorem [Cowen, GG] Suppose L is an invariant subspace forthe universal operator T∗ϕ and the block matrix(
A B0 C
)represents T∗ϕ based on the splitting H2 = M⊕M⊥. Then, the
projection of L⊥ onto M is an invariant linear manifold for A∗, theadjoint of the restriction of T∗ϕ to M.
Remark.
![Page 132: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/132.jpg)
Consequences, Further Observations, and a Question
Theorem [Cowen, GG] Suppose L is an invariant subspace forthe universal operator T∗ϕ and the block matrix(
A B0 C
)represents T∗ϕ based on the splitting H2 = M⊕M⊥. Then, the
projection of L⊥ onto M is an invariant linear manifold for A∗, theadjoint of the restriction of T∗ϕ to M.
Remark. Any of the linear manifolds provided by this Theorem areproper and invariant but, in principle, they are not necessarilynon-dense.
![Page 133: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/133.jpg)
Consequences, Further Observations, and a Question
Question: Is any of those proper A∗-invariant linear manifoldsnon-dense?
![Page 134: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/134.jpg)
Bibliography (basic)
[ChP] I. Chalendar, and J. R. Partington, Modern Approaches tothe Invariant Subspace Problem, Cambridge University Press, 2011.
[CG1] C. Cowen and E. A. Gallardo-Gutierrez, Unitary equivalenceof one-parameter groups of Toeplitz and composition operators,Journal of Functional Analysis, 261, 2641–2655, (2011).
[CG2] C. Cowen and E. A. Gallardo-Gutierrez, Rota’s universaloperators and invariant subspaces in Hilbert spaces, preprint(2014).
[CG3] C. Cowen and E. A. Gallardo-Gutierrez, Consequences ofuniversality among Toeplitz operators, Journal of MathematicalAnalysis and Applications (2015).
![Page 135: The Invariant Subspace Problem: General Operator Theory vs ... › msci › 2015prmo › pdfs › gallardo.pdf · o \On the invariant subspace problem for Banach spaces", Acta Math](https://reader033.vdocument.in/reader033/viewer/2022042316/5f04b32e7e708231d40f44f0/html5/thumbnails/135.jpg)
Bibliography (basic)
[Lo] V. Lomonosov, On invariant subspaces of families of operatorscommuting with a completely continuous operator, FunkcionalAnal. i Prilozen (1973).
[GG] E. A. Gallardo-Gutierrez and P. Gorkin, Minimal invariantsubspaces for composition operators, Journal de MathematiquesPures et Appliquees (2011).
[GGS] E. A. Gallardo-Gutierrez, P. Gorkin and D. Suarez, Orbits ofnon-elliptic disc automorphisms on Hp, Journal of MathematicalAnalysis and Applications (2012).
[NRW] E. A. Nordgren, P. Rosenthal and F. S. Wintrobe Invertiblecomposition operators on Hp, J. Functional Analysis, 73 (1987),324–344.
[Ro] G. C. Rota, On models for linear operators. Comm. PureAppl. Math. 13 (1960) 469–472.