The Invariant Subspace Problem: General Operator Theory vs. Concrete Operator Theory? ·...

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

Introduction

Introduction

r Invariant Subspace Problem

Introduction

r Invariant Subspace Problem

T : H → H, linear and bounded

T (M) ⊂ (M), closed subspace

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.

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.

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.

Example: R2

T =

(0 −11 0

),

respect to the canonical bases {e1, e2}.

Example: R2

T =

(0 −11 0

),

respect to the canonical bases {e1, e2}.

T has no non-trivial invariant subspaces in R2

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.

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.

Introduction

r `2 = {{an}n≥1 ⊂ C :∞∑n=1

|an|2 <∞}

{en}n≥1 canonical bases in `2

Introduction

r `2 = {{an}n≥1 ⊂ C :∞∑n=1

|an|2 <∞}

{en}n≥1 canonical bases in `2

Sen = en+1 n ≥ 1

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?

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.

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) = ∅.

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.

Introduction

r Classes of operators with known invariant subspaces:

Introduction

r Classes of operators with known invariant subspaces:

? Normal operators (Spectral Theorem)

Introduction

r Classes of operators with known invariant subspaces:

? Normal operators (Spectral Theorem)

? Compact Operators

Introduction

r Classes of operators with known invariant subspaces:

? Normal operators (Spectral Theorem)

? Compact Operators

σ(T ) = {λj}j≥1 ∪ {0}

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)

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)

Introduction

r Classes of operators with known invariant subspaces:

? Normal operators (Spectral Theorem)

? Compact Operators

? Polynomially Compact Operators

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)

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

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...

In the Banach space setting

In 1975, P. Enflo showed in the Seminaire Maurey-Schwartz at theEcole Polytechnique in Parıs:

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.

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.

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.

The big open question

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?

Classes of operators with known invariant subspaces

r 1973, Lomonosov

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.

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.

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.

Classes of operators with known invariant subspaces

Classes of operators with known invariant subspaces

r Does every operator satisfy “Lomonosov Hypotheses”?

Classes of operators with known invariant subspaces

r Does every operator satisfy “Lomonosov Hypotheses”?

r 1980, Hadwin; Nordgren; Radjavi y Rosenthal

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.

A “Concrete Operator Theory” approach

r Universal Operators (in the sense of G. C. Rota)

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 ,

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.

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.

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).

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.

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:

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.

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

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Examples of universal operatorsr Theorem (1987, Nordgren, Rosenthal y Wintrobe)

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.

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.

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.

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.

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.

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))

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.

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.

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.

Examples of universal operators

Every linear bounded operator T has a closednon-trivial invariant subspace

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}.

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

Question

Which conditions on f ensure that Kf := span{Cnϕf : n ≥ 0} is (or

not) minimal?

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.

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.

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.

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.

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.

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 .

Question

Eigenfuntions of Cϕ?

Question

Eigenfuntions of Cϕ?

r 2012, GG, Gorkin and Suarez, Constructive characterization ofeigenfunctions of Cϕ in the Hardy spaces Hp

Universal operators vs. Lomonosov Theorem

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?

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.

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.

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.

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.

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:

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:

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 · · ·

. . .

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,

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,

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,

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,

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,

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!

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).

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.

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.

Universal operators

Universal operators

r Theorem [2013, Cowen-GG] There exists a universal operatorwhich commutes with an injective, dense range compact operator.

A universal operator which commutes with a compact operator

Compact operators commuting with universal operators

We have seen that some universal operators commute with acompact operator and others do not.

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.

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}

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).

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}.

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).

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.

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.

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.

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

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

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.

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).

Consequences, Further Observations, and a Question

• From the relation, TW = WT, it follows that

AP + BR = PA and CR = RA

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Consequences, Further Observations, and a Question

Question: Is any of those proper A∗-invariant linear manifoldsnon-dense?

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).

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.