an extension of a theorem of domar on invariant...

PreliminaresWeighted L2 spaces

Extension of Domar’s Theorem

An extension of a Theorem of Domar on invariantsubspaces

Daniel J. Rodrıguez

Universidad de Zaragoza, Spain

Alquezar, 17-19 October 2014

Joint work with Eva A. Gallardo-Gutierrez (Madrid, Spain) andJonathan R. Partington (Leeds, UK).

Daniel J. Rodrıguez An extension of a Theorem of Domar on invariant subspaces

1 PreliminaresTranslation-invariant subspacesBeurling-Lax Theorem

2 Weighted L2 spacesDomar’s Theorem

3 Extension of Domar’s TheoremA First ApproachA General Statement

Translation-invariant subspacesBeurling-Lax Theorem

Preliminares

L2(R) = {f : R→ C measurable s.t∫∞−∞ |f (t)|2dt <∞}.

A closed subspace M⊂ L2(R) is said to be translation-invariant iff ∈M implies that fτ ∈M for every real number τ , where

fτ (t) = f (t − τ), (t ∈ R).

Problem

Describe the closed translation-invariant subspaces of L2(R).

For f ∈ L1(R) ∩ L2(R), the Fourier Transform is defined by

(F f )(x) = f (x) =

∫ ∞−∞

f (t) exp(−ixt)dt, (x ∈ R).

Preliminares

fτ (t) = f (t − τ), (t ∈ R).

Problem

(F f )(x) = f (x) =

∫ ∞−∞

Preliminares

fτ (t) = f (t − τ), (t ∈ R).

Problem

(F f )(x) = f (x) =

∫ ∞−∞

Preliminares

fτ (t) = f (t − τ), (t ∈ R).

Problem

(F f )(x) = f (x) =

∫ ∞−∞

Plancherel Theorem

f ∈ L1(R) ∩ L2(R) −→ f ∈ L2(R)

extends to an isometric isomorphism from L2(R) to L2(R).

For every measurable set E ⊂ R, define

ME = {f ∈ L2(R) : f = 0 a.e. on E}.

Theorem

For every measurable set E ⊂ R, the subspace ME is a closedtranslation-invariant subspace of L2(R). Moreover, every closedtranslation-invariant subspace of L2(R) is ME for somemeasurable set E and MA =MB if and only if

m((A \ B) ∪ (B \ A)) = 0.

Plancherel Theorem

f ∈ L1(R) ∩ L2(R) −→ f ∈ L2(R)

ME = {f ∈ L2(R) : f = 0 a.e. on E}.

Theorem

m((A \ B) ∪ (B \ A)) = 0.

Plancherel Theorem

f ∈ L1(R) ∩ L2(R) −→ f ∈ L2(R)

ME = {f ∈ L2(R) : f = 0 a.e. on E}.

Theorem

m((A \ B) ∪ (B \ A)) = 0.

Let denote R+ = {x ∈ R : x ≥ 0} and C+ = {z ∈ C : <z > 0}.

Recall:

L2(R+) = {f : R+ → C measurable s.t∫∞

0 |f (t)|2dt <∞}.

H2(C+) = {F ∈ H(C+) : supx>0

∫ ∞−∞|F (x + iy)|2dy <∞}.

The clasical Paley-Wiener Theorem states that the LaplaceTransform

(Lf )(s) =

∫ ∞0

f (t) exp(−st)dt, (s ∈ C+),

induces (up to the constant) an unitary equivalence betweenL2(R+) and H2(C+).

Recall:

0 |f (t)|2dt <∞}.

H2(C+) = {F ∈ H(C+) : supx>0

∫ ∞−∞|F (x + iy)|2dy <∞}.

(Lf )(s) =

∫ ∞0

Recall:

0 |f (t)|2dt <∞}.

H2(C+) = {F ∈ H(C+) : supx>0

∫ ∞−∞|F (x + iy)|2dy <∞}.

(Lf )(s) =

∫ ∞0

Recall:

0 |f (t)|2dt <∞}.

H2(C+) = {F ∈ H(C+) : supx>0

∫ ∞−∞|F (x + iy)|2dy <∞}.

(Lf )(s) =

∫ ∞0

Recall:

0 |f (t)|2dt <∞}.

H2(C+) = {F ∈ H(C+) : supx>0

∫ ∞−∞|F (x + iy)|2dy <∞}.

(Lf )(s) =

∫ ∞0

Beurling-Lax Theorem

Given τ ≥ 0, define the right shift operator Sτ in L2(R+) by

Note that {Sτ}τ≥0 has a rich lattice of invariant subspaces.

(Sτ f )(t) =

{0 if 0 ≤ t ≤ τ ,

f (t − τ) if t > τ,f ∈ L2(R+).

(Sτ f )(t) =

{0 if 0 ≤ t ≤ τ ,

f (t − τ) if t > τ,f ∈ L2(R+).

Problem

Describe the Lat{Sτ : τ ≥ 0} of L2(R+).

(Sτ f )(t) =

{0 if 0 ≤ t ≤ τ ,

f (t − τ) if t > τ,f ∈ L2(R+).

Problem

A non-zero closed subspace M⊂ L2(R+) satisfies SτM⊆M forall τ ≥ 0 if and only if LM = θH2(C+) for some inner functionθ ∈ H∞(C+).

(Sτ f )(t) =

{0 if 0 ≤ t ≤ τ ,

f (t − τ) if t > τ,f ∈ L2(R+).

Problem

A non-zero closed subspace M⊂ L2(R+) satisfies SτM⊆M forall τ ≥ 0 if and only if LM = θH2(C+) for some inner functionθ ∈ H∞(C+).

Extension of Domar’s TheoremDomar’s Theorem

Weighted L2 spaces

There are many generalization of the Paley-Wiener Theorem:

[Duren - Gallardo-Gutierrez - Montes-Rodrıguez]

The space L2

C (α)

πt−1

dt) corresponds to the Bergmanspace A2

The space L2α+1(R+,C (α)πt−1−αdt) corresponds to the Bergman

space A2α(C+).

space A2α(C+). [Duren - Gallardo-Gutierrez - Montes-Rodrıguez]

Bergman spaces A2α(C+) for (α > −1)

A2α(C+) = {F ∈ H(C+) :

∫∞−∞

∫∞0 |F (x + iy)|2xαdxdy <∞}.

A2α(C+) = {F ∈ H(C+) :

∫∞−∞

∫∞0 |F (x + iy)|2xαdxdy <∞}.

The space L2(R+, ω(t)dt) corresponds to the Zen space A2ν when

ω(t) = 2π∫∞

0 e−2xtd ν(x) for all t > 0.

A2α(C+) = {F ∈ H(C+) :

∫∞−∞

∫∞0 |F (x + iy)|2xαdxdy <∞}.

ω(t) = 2π∫∞

0 e−2xtd ν(x) for all t > 0. [Harper - Partington]

A2α(C+) = {F ∈ H(C+) :

∫∞−∞

∫∞0 |F (x + iy)|2xαdxdy <∞}.

ω(t) = 2π∫∞

0 e−2xtd ν(x) for all t > 0. [Harper - Partington]

Zen spaces A2ν(C+)

A2ν(C+) = {F ∈ H(C+) : supε>0

∫C+|F (z + ε)|2dν(z) <∞}, with

dν = d ν ⊗ dx where ν, ν are both positive regular Borel measureson C+ and R+, resp., and ν satisfies a doubling condition.

Let ω be a weight in R+, that is, a positive Borel function suchthat

supy∈R+

ω(x + y)

is locally bounded.

Sτ is bounded on L2(R+, ω(t)dt) for all τ ≥ 0.

Question

How is the structure of Lat{Sτ : τ ≥ 0} in L2(R+, ω(t)dt)?

For all a ∈ R+ ∪ {0} ∪ {∞}, the “standard subspaces” are

L2([a,∞), ω(t)dt) = {f ∈ L2(R+, ω(t)dt) : f (t) = 0 a.e. 0 ≤ t ≤ a}.

supy∈R+

ω(x + y)

is locally bounded. Sτ is bounded on L2(R+, ω(t)dt) for all τ ≥ 0.

Question

supy∈R+

ω(x + y)

Question

supy∈R+

ω(x + y)

Question

A more basic question

Do the spaces L2(R+, ω(t)dt) have non-standard invariantsubspaces for {Sτ : τ ≥ 0}?

Very little is known.

But, it is known is the existence of fairly large classes of weightsfor which the lattices have non-standard subspaces: classescharacterized by bounds on the size of the weight function ω and1/ω at infinity (Atzmon, Borichev-Hedenlman, Domar, Nikolskii,Thomas...)

Domar’s Theorem

Theorem (Y. Domar, 1983)

Let ω be a positive continuous function in R+ such that logω isconcave in [c ,∞) for some c ≥ 0. Assume that

1 lımt→∞

− logω(t)

t=∞.

2 lımt→∞

log | logω(t)| − log t√log t

Then all closed invariant subspaces of the right shift operators{Sτ}τ≥0 in L2(R+, ω(t)dt) are the standard.

In this case, observe that the semigroup {Sτ : τ ≥ 0} has very fewclosed invariant subspaces in L2(R+, ω(t)dt).

Domar’s Theorem

1 lımt→∞

− logω(t)

t=∞.

2 lımt→∞

Domar’s Theorem

1 lımt→∞

− logω(t)

t=∞.

2 lımt→∞

Domar’s Theorem

1 lımt→∞

− logω(t)

t=∞.

2 lımt→∞

Domar’s Theorem

1 lımt→∞

− logω(t)

t=∞.

2 lımt→∞

About Domar’s Theorem

Transfer to a problem in complex function theory.

Accurate bounds of Laplace Transform of the functionsconsidered.

Use the Ahlfors-Beurling Theorem on bounds of complexanalytic functions in an n-connected domain.

Domar’s Problem 3

How relevant is the concavity assumption on logω?.

About Domar’s Theorem

Transfer to a problem in complex function theory.

Accurate bounds of Laplace Transform of the functionsconsidered.

Use the Ahlfors-Beurling Theorem on bounds of complexanalytic functions in an n-connected domain.

Domar’s Problem 3

How relevant is the concavity assumption on logω?.

A First ApproachA General Statement

A First Approach

Recall that a sequence of positive real numbers {an}∞n=1 is said tobe logarithmically concave if a2

n ≥ an−1an+1 for all n ≥ 2.

Theorem 1 [GPR]

Let ω > 0 be a continuous decreasing function in R+ such that

(H1) {ω(n)}∞n=1 is logarithmically concave.

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

Then all closed invariant subspaces for {Sτ}τ≥0 in L2(R+, ω(t) dt)are standard.

A First Approach

Theorem 1 [GPR]

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

A First Approach

Theorem 1 [GPR]

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

A First Approach

Theorem 1 [GPR]

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

A First Approach

Theorem 1 [GPR]

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

A First Approach

Theorem 1 [GPR]

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

A First Approach

Theorem 1 [GPR]

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

Key ingredient result

Proposition

Let ω > 0 be a continuous decreasing function in R+. Let M be anon-trivial invariant subspace for {Sτ}τ≥0 in L2(R+, ω(t)dt).Assume M contains a non-trivial standard invariant subspace for{Sτ}τ≥0. Then M is standard.

Key idea: Construct a positive continuous function ωa in R+

(related to ω) which satisfies the hypotheses of Domar’s Theorem.

Key ingredient result

Proposition

Let ω > 0 be a continuous decreasing function in R+. Let M be anon-trivial invariant subspace for {Sτ}τ≥0 in L2(R+, ω(t)dt).Assume M contains a non-trivial standard invariant subspace for{Sτ}τ≥0. Then M is standard.

Key idea: Construct a positive continuous function ωa in R+

(related to ω) which satisfies the hypotheses of Domar’s Theorem.

Proof of Theorem 1

WLOG, we may assume that 0 < ω(t) ≤ 1 for all t ∈ R+.

Under hypothesis (H1) - (H3), we are able to construct a positivefunction ωa in R+ satisfying

(C1) ω(t) ≤ ωa(t), for all t ∈ R+.

(C2) ωa is log-concave in R+.

(C3) lımt→∞

− logωa(t)

t=∞.

(C4) lımt→∞

log | logωa(t)| − log t√log t

Proof of Theorem 1

(C3) lımt→∞

− logωa(t)

t=∞.

(C4) lımt→∞

Proof of Theorem 1

(C3) lımt→∞

− logωa(t)

t=∞.

(C4) lımt→∞

Proof of Theorem 1

(C3) lımt→∞

− logωa(t)

t=∞.

(C4) lımt→∞

Proof of Theorem 1

(C3) lımt→∞

− logωa(t)

t=∞.

(C4) lımt→∞

Proof of Theorem 1

(C3) lımt→∞

− logωa(t)

t=∞.

(C4) lımt→∞

Proof of Theorem 1

(C3) lımt→∞

− logωa(t)

t=∞.

(C4) lımt→∞

For each n ≥ 0, let Λn be the linear function defined in the interval[n, n + 1) by

Λn(t) = logω(n)+(t−n) (logω(n + 1)− logω(n)) , (n ≤ t < n+1).

(n, logω(n))

(n + 1, logω(n + 1))

Consider the piecewise function

Λ(t) =∑n≥0

Λn(t)χ[n,n+1)(t), (t ∈ R+)

which is concave for t ≥ 1.

(n, logω(n))

(n + 1, logω(n + 1))

Λ(t) =∑n≥0

Λn(t)χ[n,n+1)(t), (t ∈ R+)

(n, logω(n))

(n + 1, logω(n + 1))

Λ(t) =∑n≥0

Λn(t)χ[n,n+1)(t), (t ∈ R+)

(n, logω(n))

(n + 1, logω(n + 1))

Λ(t) =∑n≥0

Λn(t)χ[n,n+1)(t), (t ∈ R+)

(n, logω(n))

(n + 1, logω(n + 1))

Λ(t) =∑n≥0

Λn(t)χ[n,n+1)(t), (t ∈ R+)

(n, logω(n))

(n + 1, logω(n + 1))

Λ(t) =∑n≥0

Λn(t)χ[n,n+1)(t), (t ∈ R+)

which is concave for t ≥ 1.Daniel J. Rodrıguez An extension of a Theorem of Domar on invariant subspaces

Let us define the positive function ωa as follows

ωa(t) =

{ω(0) if 0 ≤ t < 1,exp(Λ(t − 1)) if t ≥ 1.

0 1 2 3 t

logω(1)

logω(2)

logω(3)

ΛΛ(t − 1)

ωa(t) =

{ω(0) if 0 ≤ t < 1,exp(Λ(t − 1)) if t ≥ 1.

0 1 2 3 t

logω(1)

logω(2)

logω(3)

ΛΛ(t − 1)

ωa(t) =

{ω(0) if 0 ≤ t < 1,exp(Λ(t − 1)) if t ≥ 1.

0 1 2 3 t

logω(1)

logω(2)

logω(3)

ΛΛ(t − 1)

ωa(t) =

{ω(0) if 0 ≤ t < 1,exp(Λ(t − 1)) if t ≥ 1.

0 1 2 3 t

logω(1)

logω(2)

logω(3)Λ

Λ(t − 1)

ωa(t) =

{ω(0) if 0 ≤ t < 1,exp(Λ(t − 1)) if t ≥ 1.

0 1 2 3 t

logω(1)

logω(2)

logω(3)Λ

Λ(t − 1)

By construction:

(C1) ω(t) ≤ ωa(t), for all t ∈ R+. (C2) logωa is concave in R+.

By construction:

(C2) logωa is concave in R+.

By construction:

Moreover, since ωa(t + 2) ≤ ω(t) for all t ≥ 1, conditions (C3)and (C4) follows from the fact that ω satisfies (H2) and (H3).

By construction:

Hypotheses

(H2) lımt→∞

− logω(t)

t=∞. (H3) lım

t→∞

By construction:

Now, condition (C1) implies L2(R+, ωa(t)dt) ⊂ L2(R+, ω(t)dt).

By construction:

Let M⊂ L2(R+, ω(t) dt) be a closed, non-trivial invariantsubspace for {Sτ}τ≥0.

By construction:

Let M⊂ L2(R+, ω(t) dt) be a closed, non-trivial invariantsubspace for {Sτ}τ≥0. Observe that S2M⊂ L2(R+, ωa(t)dt).

By construction:

Since S2Mωa ⊂ L2(R+, ωa(t)dt) is a closed invariant subspace for

{Sτ}τ≥0, by Domar’s Theorem

S2Mωa

= L2([ca,∞), ωa(t) dt), for some ca ∈ R+ ∪ {0} ∪ {∞}.

By construction:

Since S2Mωa ⊂ L2(R+, ωa(t)dt) is a closed invariant subspace for

{Sτ}τ≥0, by Domar’s Theorem

S2Mωa

= L2([ca,∞), ωa(t) dt), for some ca ∈ R+ ∪ {0} ∪ {∞}.

Observe that ca <∞, since M 6= {0}.

Using a density argument and the chain S2Mωa ⊂ S2M

ω ⊂M, itfollows that

L2([ca,∞), ω(t) dt) ⊆M.

In addition, ca > 0 since M is proper.

By the previous proposition, M is standard, which concludes theproof.

Remarks

1 Hypothesis (H1) can be replaced by (H1’) {ω(n + k)}∞n=1 islogarithmically concave for some positive number k .

2 The inclusion L2(R+, ωa(t) dt) ⊂ L2(R+, ω(t) dt) is proper,since for every t ∈ [n, n + 1), n ≥ 2, one has

ωa(t)≤ ω(n)

ωa(n)=

(ω(n)

ω(n − 1)

)n+1−t.

L2([ca,∞), ω(t) dt) ⊆M.

Remarks

ωa(t)≤ ω(n)

ωa(n)=

(ω(n)

ω(n − 1)

)n+1−t.

L2([ca,∞), ω(t) dt) ⊆M.

Remarks

ωa(t)≤ ω(n)

ωa(n)=

(ω(n)

ω(n − 1)

)n+1−t.

L2([ca,∞), ω(t) dt) ⊆M.

Remarks

ωa(t)≤ ω(n)

ωa(n)=

(ω(n)

ω(n − 1)

)n+1−t.

L2([ca,∞), ω(t) dt) ⊆M.

Remarks

ωa(t)≤ ω(n)

ωa(n)=

(ω(n)

ω(n − 1)

)n+1−t.

L2([ca,∞), ω(t) dt) ⊆M.

Remarks

ωa(t)≤ ω(n)

ωa(n)=

(ω(n)

ω(n − 1)

)n+1−t.

Observe that condition (H1) plays a fundamental role in the proof.

Hypothesis

(H1) {ω(n)}∞n=1 is logarithmically concave

Hypothesis

(H1) {ω(n)}∞n=1 is logarithmically concave if and only if thesequence {logω(n + 1)− logω(n)}∞n=1 is decreasing.

For non-equidistant points, one finds that the interpolation methodno longer applies.

For instance , fixed 0 < ε < 1, define the increasing sequence

t2n−1 = 2n − 1 , t2n = t2n−1 + ε, (n ≥ 1).

Consider ω > 0 the continuous decreasing function such that

logω(0) = 0 and logω(tn) = −n (n ≥ 1).

t2n−1 = 2n − 1 , t2n = t2n−1 + ε, (n ≥ 1).

If Λ denotes the polygonal function, we see that

t2n−1 = 2n − 1 , t2n = t2n−1 + ε, (n ≥ 1).

• the sequence {ω(tn)}∞n=1 is logarithmically concave but

t2n−1 = 2n − 1 , t2n = t2n−1 + ε, (n ≥ 1).

• the sequence {ω(tn)}∞n=1 is logarithmically concave but• the function W (t) = exp(Λ(t)) is not log-concave in R+.

0 t1 2 t3t

A General Statement

Theorem 2 [GPR]

Let ω > 0 be a continuous decreasing function in R+ such that:

(H1) There exists a strictly increasing sequence {tn}∞n=1 ⊂ R+

with tn →∞ as n→∞ such that supn(tn+1 − tn) <∞ and{(ω(tn+1)/ω(tn))1/(tn+1−tn)}∞n=1 is a decreasing sequence.

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

A General Statement

Theorem 2 [GPR]

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

A General Statement

Theorem 2 [GPR]

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

A General Statement

Theorem 2 [GPR]

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

A General Statement

Theorem 2 [GPR]

(H2) lımt→∞

− logω(t)

t=∞.

(H3) lımt→∞

Remark

If {tn}∞n=1 satisfies (H1), then the sequence

{(ω(tn+1)/ω(tn))1/(tn+1−tn)}∞n=1

is decreasing. Therefore, we recover the logarithmically concavecondition (H1) of Theorem 1 whenever {tn}∞n=1 consists ofequidistant points in R+.

Remark

If {tn}∞n=1 satisfies (H1), then the sequence

{(ω(tn+1)/ω(tn))1/(tn+1−tn)}∞n=1

is decreasing. Therefore, we recover the logarithmically concavecondition (H1) of Theorem 1 whenever {tn}∞n=1 consists ofequidistant points in R+.

Sketch of the proof of Theorem 2

1 Assume 0 < ω(t) ≤ 1,∀t ∈ R+ and M = supn(tn+1 − tn) ≤ 1.

2 For each n ≥ 1, construct Λn be the polygonal function.

3 Consider the concave linear piecewise function Λ.

4 Define the positive function

ωa(t) =