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Radial Toeplitz operators on the Bergman space and slowly oscillating sequences

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Radial Toeplitz operators on the Bergman space

and slowly oscillating sequences

Egor Maximenko

Instituto Polit´ecnico Nacional, ESFM, M´exico

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The results are joint with

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Contents

Bergman space and the monomial basis

Toeplitz operators with radial generating symbols

Description of RT found by Daniel Su´arez

Slowly oscillating sequences

Main result: RT ∼= SO1

(4)

Contents

Bergman space and the monomial basis

Toeplitz operators with radial generating symbols

Description of RT found by Daniel Su´arez

Slowly oscillating sequences

Main result: RT ∼= SO1

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Bergman space

D := {z ∈ C : |z | < 1} .

µ := plane Lebesgue measure.

A2(D) = Bergman space := {f ∈ L2(D, µ) : f is analitic}.

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Monomial basis, Bergman projection

The normalized monomials form an orthonormal basis of A2(D):

ϕn(z) := q n+1 π z n. Bergman kernel: K (z, w ) = ∞ X n=0 ϕn(z)ϕn(w ) = 1 π(1 − zw )2. Bergman projection : (Bf )(z) := ∞ X n=0 hf , ϕni ϕn= Z D K (z, w ) f (w ) d µ(w ).

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Isometric isomorphism between A

2

(D) and `

2

Since A2(D) is an infinite-dimensional separable Hilbert space,

A2(D) ∼= `2,

where `2 := the space of quadratically summable complex sequences.

Using the monomial basis (ϕn)∞n=0 of A2(D)

define the following isometric isomorphism

R : A2(D) → `2. R : f ∈ A2(D) 7→ (hf , ϕni)∞n=0∈ `2. R−1: (xn)∞n=0∈ `2 7→ ∞ X n=0 xnϕn∈ A2(D).

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Isometric isomorphism between A

2

(D) and `

2

Since A2(D) is an infinite-dimensional separable Hilbert space,

A2(D) ∼= `2,

where `2 := the space of quadratically summable complex sequences. Using the monomial basis (ϕn)∞n=0 of A2(D)

define the following isometric isomorphism

R : A2(D) → `2. R : f ∈ A2(D) 7→ (hf , ϕni)∞n=0∈ `2. R−1: (xn)∞n=0∈ `2 7→ ∞ X n=0 xnϕn∈ A2(D).

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Contents

Bergman space and the monomial basis

Toeplitz operators with radial generating symbols

Description of RT found by Daniel Su´arez

Slowly oscillating sequences

Main result: RT ∼= SO1

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Radial Toeplitz operators on the Bergman space

Given a function a ∈ L∞(D),

the Toeplitz operator with defining symbol a is

Ta: A2(D) → A2(D), Taf = B(af ).

Here we consider only “radial Toeplitz operators”, i.e. the Toeplitz operators with radial defining symbols .

A function a : D → C is called radial if

a(z) = a(|z|) ∀z ∈ D.

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Radial Toeplitz operators on the Bergman space

Given a function a ∈ L∞(D),

the Toeplitz operator with defining symbol a is

Ta: A2(D) → A2(D), Taf = B(af ).

Here we consider only “radial Toeplitz operators”, i.e. the Toeplitz operators with radial defining symbols .

A function a : D → C is called radial if

a(z) = a(|z|) ∀z ∈ D.

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Radial Toeplitz operators on the Bergman space

Given a function a ∈ L∞(D),

the Toeplitz operator with defining symbol a is

Ta: A2(D) → A2(D), Taf = B(af ).

Here we consider only “radial Toeplitz operators”, i.e. the Toeplitz operators with radial defining symbols .

A function a : D → C is called radial if

a(z) = a(|z|) ∀z ∈ D.

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Algebra generated by radial Toeplitz operators

Consider the set of all Toeplitz operators with bounded radial defining symbols:

Λ := {Ta: a ∈ L(0, 1)}.

The object of this work is the C∗-algebra generated by these operators: RT := C-algebra generated by Λ.

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Diagonalization of radial Toeplitz operators

Theorem (Korenblum and Zhu, 1995)

Let a ∈ L(D) be a radial function. Then

RTaR−1 = Mγa, where Mγa: `

2 → `2 is the multiplication operator

by the following sequence γa ∈ `:

γa(n) = (n + 1)

Z 1 0

a(r )rndr .

In other words, the monomial basis (ϕn)∞n=0 diagonalizes Ta,

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Diagonalization of radial Toeplitz operators

Theorem (Korenblum and Zhu, 1995)

Let a ∈ L(D) be a radial function. Then

RTaR−1 = Mγa, where Mγa: `

2 → `2 is the multiplication operator

by the following sequence γa ∈ `:

γa(n) = (n + 1)

Z 1 0

a(r )rndr .

In other words, the monomial basis (ϕn)∞n=0 diagonalizes Ta,

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Proof of the theorem about the diagonalization

hTaϕn, ϕki = hB(aϕn), ϕki = haϕn, B(ϕk)i = haϕn, ϕki

= n + 1 π Z D a(z)znzkd µ(z) = n + 1 π Z 0 ei (n−k)ϕd ϕ Z 1 0 a(r ) rn+k+1dr = δn,k · (n + 1) Z 1 0 a(u)undu = δn,k · γa(n). From here Taϕn= γa(n)ϕn and RTaR−1x = γax ∀x ∈ `2.

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Proof of the theorem about the diagonalization

hTaϕn, ϕki = hB(aϕn), ϕki = haϕn, B(ϕk)i = haϕn, ϕki

= n + 1 π Z D a(z)znzkd µ(z) = n + 1 π Z 0 ei (n−k)ϕd ϕ Z 1 0 a(r ) rn+k+1dr = δn,k · (n + 1) Z 1 0 a(u)undu = δn,k · γa(n). From here Taϕn= γa(n)ϕn and RTaR−1x = γax ∀x ∈ `2.

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Passing from

operators

to

sequences

Ta γa

Λ := {Ta: a ∈ L(0, 1)} Γ := {γa: a ∈ L(0, 1)}

RT := C-algebra(Λ) A := C-algebra(Γ)

The mappings Ta7→ Mγa 7→ γa are linear, multiplicative and isometric.

The C∗-algebras RT and A are isometrically isomorphic.

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Passing from

operators

to

sequences

Ta γa

Λ := {Ta: a ∈ L(0, 1)} Γ := {γa: a ∈ L(0, 1)}

RT := C-algebra(Λ) A := C-algebra(Γ)

The mappings Ta7→ Mγa 7→ γa are linear, multiplicative and isometric.

The C∗-algebras RT and A are isometrically isomorphic.

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Passing from

operators

to

sequences

Ta γa

Λ := {Ta: a ∈ L(0, 1)} Γ := {γa: a ∈ L(0, 1)}

RT := C-algebra(Λ) A := C-algebra(Γ)

The mappings Ta7→ Mγa 7→ γa are linear, multiplicative and isometric.

The C∗-algebras RT and A are isometrically isomorphic.

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Contents

Bergman space and the monomial basis

Toeplitz operators with radial generating symbols

Description of RT found by Daniel Su´arez

Slowly oscillating sequences

Main result: RT ∼= SO1

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Iterated differences of a sequence

Given a sequence x = (xn)∞n=0, define its differences of the first order:

(∆x )n= xn+1− xn,

the differences of the second order:

(∆2x )n= (∆x )n+1− (∆x )n= xn+2− 2xn+1+ xn,

etc.

Differences of the order k:

(∆kx )n= (−1)k k X j=0 (−1)j k j ! xn+j.

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A classical result from the moment theory

Given a function a : [0, 1] → C, its moment sequence is (µa(n))n=0,

µa(n) :=

Z 1 0

a(t)tndt.

Criterion of moment sequence of a bounded function (based on results of Hausdorff)

Given a sequence (xn)∞n=0, the following conditions are equivalent:

(xn)∞n=0 is the moment sequence of a bounded function;

sup k,n≥0 (n + k + 1) n + k k ! (∆kx )n < +∞.

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A classical result from the moment theory

Given a function a : [0, 1] → C, its moment sequence is (µa(n))n=0,

µa(n) :=

Z 1 0

a(t)tndt.

Criterion of moment sequence of a bounded function (based on results of Hausdorff)

Given a sequence (xn)∞n=0, the following conditions are equivalent:

(xn)∞n=0 is the moment sequence of a bounded function;

sup k,n≥0 (n + k + 1) n + k k ! (∆kx )n < +∞.

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Description of Γ through iterated differences

Recall that Γ := {γa: a ∈ L(0, 1)}, where

γa(n) = (n + 1)

Z 1 0

a(r )rndr .

Compare to the definition of the moment sequence:

µa(n) :=

Z 1 0

a(t)tndt.

Description of Γ through iterated differences (Daniel Su´arez, 2008)

A sequence x = (xn)∞n=0 belongs to Γ ⇐⇒ sup k,n≥0 (n + k + 1) n + k k ! (∆ky )n < +∞ where yn= xn n + 1.

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Description of A by Su´

arez

Theorem (Daniel Su´arez, 2005)

The set Γ not only generates the C-algebra A, but is also dense in A:

A is the closure of Γ in `.

Let d1 be the set of all bounded sequences (xn)∞n=0 such that

sup n≥0  (n + 1)|xn+1− xn|  < +∞.

Theorem (Daniel Su´arez, 2008)

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Description of A by Su´

arez

Theorem (Daniel Su´arez, 2005)

The set Γ not only generates the C-algebra A, but is also dense in A:

A is the closure of Γ in `.

Let d1 be the set of all bounded sequences (xn)∞n=0 such that

sup n≥0  (n + 1)|xn+1− xn|  < +∞.

Theorem (Daniel Su´arez, 2008)

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Description of A by Su´

arez

Theorem (Daniel Su´arez, 2005)

The set Γ not only generates the C-algebra A, but is also dense in A:

A is the closure of Γ in `.

Let d1 be the set of all bounded sequences (xn)∞n=0 such that

sup n≥0  (n + 1)|xn+1− xn|  < +∞.

Theorem (Daniel Su´arez, 2008)

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Contents

Bergman space and the monomial basis

Toeplitz operators with radial generating symbols

Description of RT found by Daniel Su´arez

Slowly oscillating sequences

Main result: RT ∼= SO1

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The algebra SO

1

SO1 := the set of all bounded sequences x = (xn)∞n=0 such that

lim m n→1 |xm− xn| = 0, that is, ∀ε > 0 ∃δ > 0 ∀m, n  m n − 1 < δ|xm− xn| < ε  . Observation SO1 is a C-subalgebra of `∞.

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Examples of sequences in SO

1 Sequences that have a finite limit

c ( SO1

Example of a sequence x ∈ SO1 such that @ lim

n→∞xn xn= cos  6 log(n + 1)  40 1 −1 n

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Examples of sequences in SO

1 Sequences that have a finite limit

c ( SO1

Example of a sequence x ∈ SO1 such that @ limn→∞xn

xn= cos  6 log(n + 1)  40 1 −1 n

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Comparison with the algebra SO

SO consists of the bounded sequences x = (xn)∞n=0 such that

lim

n→∞|xn+1− xn| = 0.

Proposition

SO1 ( SO.

Idea of the proof

If n → ∞ and m = n + 1, then mn → 1. Therefore SO1 ⊆ SO.

To show that SO16= SO, consider the sequence

xn= cos(π

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Comparison with the algebra SO

SO consists of the bounded sequences x = (xn)∞n=0 such that

lim

n→∞|xn+1− xn| = 0.

Proposition

SO1 ( SO.

Idea of the proof

If n → ∞ and m = n + 1, then mn → 1. Therefore SO1 ⊆ SO.

To show that SO16= SO, consider the sequence

xn= cos(π

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Comparison with the algebra SO

SO consists of the bounded sequences x = (xn)∞n=0 such that

lim

n→∞|xn+1− xn| = 0.

Proposition

SO1 ( SO.

Idea of the proof

If n → ∞ and m = n + 1, then mn → 1. Therefore SO1 ⊆ SO. To show that SO16= SO, consider the sequence

xn= cos(π

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Example of a sequence ∈ SO \ SO

1 xn= cos πn 40 1 −1 n

This sequence belongs to SO:

|xn+1− xn| → 0 as n → ∞,

but doesn’t belong to SO1:

|xm− xn| 6→ 0 as m

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Contents

Bergman space and the monomial basis

Toeplitz operators with radial generating symbols

Description of RT found by Daniel Su´arez

Slowly oscillating sequences

Main result: RT ∼= SO1

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Main result

Theorem

A = SO1.

As a corollary,

RT ∼= SO1.

Idea of the proof

The result of Su´arez says that d1 is dense in A.

We prove directly that d1 is dense in SO1. So,

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