the invariant subspace problem: general operator theory vs. concrete operator theory? ·...

$: The Invariant Subspace Problem: General Operator Theory vs. Concrete Operator Theory? · 2019-08-16 · A \Concrete Operator Theory" approach: universal operators r Proposition. Let$
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


T : H → H, linear and bounded

T (M) ⊂ (M), closed subspace

Introduction




=⇒ M = {0} or M = H?

1 Finite dimensional complex Hilbert spaces.

Introduction




=⇒ M = {0} or M = H?

r Remarks


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




=⇒ M = {0} or M = H?

r Remarks


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 <∞}


Sen = en+1 n ≥ 1

Introduction

r `2 = {{an}n≥1 ⊂ C :∞∑n=1

|an|2 <∞}


Sen = en+1 n ≥ 1

Characterization of the invariant subspaces of S?

Introduction

r `2 = {{an}n≥1 ⊂ C :∞∑n=1

|an|2 <∞}


Sen = en+1 n ≥ 1


ker(S − λI ) = {0} for any λ ∈ C.

Introduction

r `2 = {{an}n≥1 ⊂ C :∞∑n=1

|an|2 <∞}


Sen = en+1 n ≥ 1


ker(S − λI ) = {0} for any λ ∈ C. That is, σp(S) = ∅.

Introduction

r `2 = {{an}n≥1 ⊂ C :∞∑n=1

|an|2 <∞}


Sen = en+1 n ≥ 1


Classical Beurling Theory:Inner-Outer Factorization of the functions in the Hardy space.

Introduction

r Classes of operators with known invariant subspaces:

Introduction


? Normal operators (Spectral Theorem)

Introduction



? Compact Operators

Introduction



? Compact Operators

σ(T ) = {λj}j≥1 ∪ {0}

Introduction



? Compact Operators

σ(T ) = {λj}j≥1 ∪ {0}

∗ 1951, J. von Newman, (Hilbert space case)

Introduction



? Compact Operators

σ(T ) = {λj}j≥1 ∪ {0}

∗ 1951, J. von Newman, (Hilbert space case)

∗ 1954, Aronszajn and Smith (general case)

Introduction



? Compact Operators

? Polynomially Compact Operators

Introduction



? Compact Operators


∗ 1966, Bernstein and Robinson, (Hilbert space case)

Introduction



? Compact Operators



∗ 1967, Halmos

Introduction



? Compact Operators



∗ 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:



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.





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


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.


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.


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.


r Does every operator satisfy “Lomonosov Hypotheses”?



r 1980, Hadwin; Nordgren; Radjavi y Rosenthal



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 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 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.




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. 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:



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




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



1 Ker(U) is infinite dimensional,

2 U is surjective.




r 1969, Caradus


2 U is surjective.




r 1969, Caradus

Let U be a linear bounded operator on a Hilbert space. Assumethat


2 U is surjective.




r 1969, Caradus

Let U be a linear bounded operator on a Hilbert space. Assumethat


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,




Idea of the Proof

Write K =Ker U


1 UV = Id,

2 UW = 0,


Step 2: Prove that U is universal.



Idea of the Proof

Write K =Ker U


1 UV = Id,

2 UW = 0,


Step 2: Prove that U is universal.Let T be a linear bounded operator on H.



Idea of the Proof

Write K =Ker U


1 UV = Id,

2 UW = 0,


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.



Idea of the Proof

Write K =Ker U


1 UV = Id,

2 UW = 0,



J =∞∑k=0

λkVkWTk.

J satisfies J = W + λVJT and therefore, UJ = λJT.



Idea of the Proof

Write K =Ker U


1 UV = Id,

2 UW = 0,



J =∞∑k=0

λkVkWTk.

J satisfies J = W + λVJT and therefore, UJ = λJT. In addition,



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.




ϕ(z) = eiθp− z

1− pz(z ∈ D).


? Parabolic.

? Hyperbolic.

? Elliptic.




ϕ(z) = eiθp− z

1− pz(z ∈ D).


? Parabolic. ϕ has just one fixed point α ∈ ∂D (⇔ |p| = cos(θ/2))

? Hyperbolic.

? Elliptic.




ϕ(z) = eiθp− z

1− pz(z ∈ D).



? Hyperbolic. ϕ has two fixed points α and β, such thatα, β ∈ ∂D (⇔ |p| > cos(θ/2))

? Elliptic.




ϕ(z) = eiθp− z

1− pz(z ∈ D).



? 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.





We may assume that ϕ fixes 1 and −1.





We may assume that ϕ fixes 1 and −1. Then,

ϕ(z) =z + r

1 + rz, 0 < r < 1.


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

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.


1 lımz→−1

|f(z)| <∞,

2 |f(z)| ≤ C|z− 1|ε for some constant C, and ε > 0 in aneighborhood of 1.


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.


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.


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.


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.


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.



• 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∗ 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,


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:



• So, the only compact operator that commutes with theuniversal operator S∗ is 0,


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:



• 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).



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 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.



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}




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).


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}.


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).


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.



• 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.






• 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).


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

AP + BR = PA and CR = RA


• 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.


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.


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.


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.



A B0 C



Remark.



A B0 C



Remark. Any of the linear manifolds provided by this Theorem areproper and invariant but, in principle, they are not necessarilynon-dense.


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.

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