[PPT] - New results on semigroups of analytic functions OSCAR BLASCO PowerPoint Presentation

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New results on semigroups of analytic functions

OSCAR BLASCO

Departamento An´ alisis Matem´ atico Universidad Valencia

2013 AHA Granada, 23 May 2013 www.uv.es/oblasco

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Semigroups of analytic functions A (one-parameter) semigroup of analytic functions is any continuous homomorphism Φ : (R+,+) → {f ∈ H∞(D) : f ∞ ≤ 1}, that is t → Φ(t) = ϕt from the additive semigroup of nonnegative real numbers into the composition semigroup of all analytic functions which map D into D.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Semigroups of analytic functions A (one-parameter) semigroup of analytic functions is any continuous homomorphism Φ : (R+,+) → {f ∈ H∞(D) : f ∞ ≤ 1}, that is t → Φ(t) = ϕt from the additive semigroup of nonnegative real numbers into the composition semigroup of all analytic functions which map D into D. Φ = (ϕt) consists of ϕt ∈ H (D) with ϕt(D) ⊂ D and satisfying

1

ϕ0 is the identity in D,

2

ϕt+s = ϕt ◦ϕs, for all t,s ≥ 0,

3

ϕt(z) → ϕ0(z) = z, as t → 0, z ∈ D.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Semigroups of analytic functions A (one-parameter) semigroup of analytic functions is any continuous homomorphism Φ : (R+,+) → {f ∈ H∞(D) : f ∞ ≤ 1}, that is t → Φ(t) = ϕt from the additive semigroup of nonnegative real numbers into the composition semigroup of all analytic functions which map D into D. Φ = (ϕt) consists of ϕt ∈ H (D) with ϕt(D) ⊂ D and satisfying

1

ϕ0 is the identity in D,

2

ϕt+s = ϕt ◦ϕs, for all t,s ≥ 0,

3

ϕt(z) → ϕ0(z) = z, as t → 0, z ∈ D. Examples: φt(z) = e−tz (Dilation semigroup) φt(z) = eitz (Rotation semigroup) φt(z) = e−tz +(1−e−t)

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of (ϕt) is the function G(z) := l´ ım

t→0+

ϕt(z)−z t = ∂ϕt ∂t (z)|t=0, z ∈ D.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of (ϕt) is the function G(z) := l´ ım

t→0+

ϕt(z)−z t = ∂ϕt ∂t (z)|t=0, z ∈ D. G(ϕt(z)) = ∂ϕt(z) ∂t = G(z)∂ϕt(z) ∂z , z ∈ D, t ≥ 0. (1)

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of (ϕt) is the function G(z) := l´ ım

t→0+

ϕt(z)−z t = ∂ϕt ∂t (z)|t=0, z ∈ D. G(ϕt(z)) = ∂ϕt(z) ∂t = G(z)∂ϕt(z) ∂z , z ∈ D, t ≥ 0. (1) G has a unique representation G(z) = (bz −1)(z −b)P(z), z ∈ D where b ∈ D ( called the Denjoy-Wolff point of the semigroup) and P ∈ H (D) with ReP(z) ≥ 0 for all z ∈ D.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of (ϕt) is the function G(z) := l´ ım

t→0+

ϕt(z)−z t = ∂ϕt ∂t (z)|t=0, z ∈ D. G(ϕt(z)) = ∂ϕt(z) ∂t = G(z)∂ϕt(z) ∂z , z ∈ D, t ≥ 0. (1) G has a unique representation G(z) = (bz −1)(z −b)P(z), z ∈ D where b ∈ D ( called the Denjoy-Wolff point of the semigroup) and P ∈ H (D) with ReP(z) ≥ 0 for all z ∈ D. G(z) = −z for the dilation semigroup (b = 0, P(z) = 1) G(z) = iz for the rotation semigroup (b = 0, P(z) = −i) G(z) = −(z −1) for φt(z) = e−tz +1−e−t (b = 1, P(z) =

1 1−z )

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Semigroups of operators Each semigroup of analytic functions gives rise to a semigroup (Ct) consisting of composition operators on H (D) via composition Ct(f ) := f ◦ϕt, f ∈ H (D).

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Semigroups of operators Each semigroup of analytic functions gives rise to a semigroup (Ct) consisting of composition operators on H (D) via composition Ct(f ) := f ◦ϕt, f ∈ H (D). Given a Banach space X ⊂ H (D) and a semigroup (ϕt), we say that (ϕt) generates a semigroup of operators on X if (Ct) is a C0-semigroup

f bounded operators in X, i.e.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Semigroups of operators Each semigroup of analytic functions gives rise to a semigroup (Ct) consisting of composition operators on H (D) via composition Ct(f ) := f ◦ϕt, f ∈ H (D). Given a Banach space X ⊂ H (D) and a semigroup (ϕt), we say that (ϕt) generates a semigroup of operators on X if (Ct) is a C0-semigroup

f bounded operators in X, i.e.

Ct(f ) ∈ X for all t ≥ 0 and for every f ∈ X l´ ımt→0+ Ct(f )−f X = 0.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Semigroups of operators Each semigroup of analytic functions gives rise to a semigroup (Ct) consisting of composition operators on H (D) via composition Ct(f ) := f ◦ϕt, f ∈ H (D). Given a Banach space X ⊂ H (D) and a semigroup (ϕt), we say that (ϕt) generates a semigroup of operators on X if (Ct) is a C0-semigroup

f bounded operators in X, i.e.

Ct(f ) ∈ X for all t ≥ 0 and for every f ∈ X l´ ımt→0+ Ct(f )−f X = 0. Given a semigroup (ϕt) and a Banach space X contained in H (D) we denote by [ϕt,X] the maximal closed linear subspace of X such that (ϕt) generates a semigroups of operators on it.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Previous results on semigroups of analytic functions Theorem

1

Every semigroup of analytic functions generates a semigroup of

perators on the Hardy spaces Hp (1 ≤ p < ∞), the Bergman spaces

Ap (1 ≤ p < ∞) and the Dirichlet space, i.e. [ϕt,X] = X in these cases.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Previous results on semigroups of analytic functions Theorem

1

Every semigroup of analytic functions generates a semigroup of

perators on the Hardy spaces Hp (1 ≤ p < ∞), the Bergman spaces

Ap (1 ≤ p < ∞) and the Dirichlet space, i.e. [ϕt,X] = X in these cases.

2

No non-trivial semigroup generates a semigroup of operators in the space H∞ of bounded analytic functions, i.e. [ϕt,H∞] = H∞ implies Φ = 0.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Previous results on semigroups of analytic functions Theorem

1

Every semigroup of analytic functions generates a semigroup of

perators on the Hardy spaces Hp (1 ≤ p < ∞), the Bergman spaces

Ap (1 ≤ p < ∞) and the Dirichlet space, i.e. [ϕt,X] = X in these cases.

2

No non-trivial semigroup generates a semigroup of operators in the space H∞ of bounded analytic functions, i.e. [ϕt,H∞] = H∞ implies Φ = 0.

3

There are plenty of semigroups (but not all) which generate semigroups of operators in the disk algebra.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The case X = BMOA Definition An analytic function f is said to belong to BMOA if f 2

∗ = sup I

1 |I|

R(I) |f ′(z)|2(1−|z|2)dA(z) < ∞

where the sup is taken over all arcs I ⊂ ∂D, R(I) is the Carleson rectangle determined by I, |I| denotes the normalized length of I and dA(z) the normalized Lebesgue measure on ∂D.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The case X = BMOA Definition An analytic function f is said to belong to BMOA if f 2

∗ = sup I

1 |I|

R(I) |f ′(z)|2(1−|z|2)dA(z) < ∞

where the sup is taken over all arcs I ⊂ ∂D, R(I) is the Carleson rectangle determined by I, |I| denotes the normalized length of I and dA(z) the normalized Lebesgue measure on ∂D. VMOA is the subspace of functions satisfying l´ ım

|I|→0

1 |I|

R(I) |f ′(z)|2(1−|z|2)dA(z) = 0

It is known that VMOA is the closure of the polynomials in BMOA and that (VMOA)∗∗ = BMOA.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The problem for BMOA Here it is our starting motivation: Theorem A. (Sarason) Suppose f ∈ BMOA, then the following are equivalent:

1

f ∈ VMOA.

2

l´ ımt→0+ f (eit·)−f ⋆ = 0.

3

l´ ımr→1 f (r·)−f ⋆ = 0.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The problem for BMOA Here it is our starting motivation: Theorem A. (Sarason) Suppose f ∈ BMOA, then the following are equivalent:

1

f ∈ VMOA.

2

l´ ımt→0+ f (eit·)−f ⋆ = 0.

3

l´ ımr→1 f (r·)−f ⋆ = 0. Note that l´ ımt→0+ f (eit·)−f ⋆ = 0 means f ∈ [eitz,BMO].

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The problem for BMOA Here it is our starting motivation: Theorem A. (Sarason) Suppose f ∈ BMOA, then the following are equivalent:

1

f ∈ VMOA.

2

l´ ımt→0+ f (eit·)−f ⋆ = 0.

3

l´ ımr→1 f (r·)−f ⋆ = 0. Note that l´ ımt→0+ f (eit·)−f ⋆ = 0 means f ∈ [eitz,BMO]. Note that l´ ımr→1 f (r·)−f ⋆ = 0 can be written l´ ımt→0+ f (e−t·)−f ⋆ = 0

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The problem for BMOA Here it is our starting motivation: Theorem A. (Sarason) Suppose f ∈ BMOA, then the following are equivalent:

1

f ∈ VMOA.

2

l´ ımt→0+ f (eit·)−f ⋆ = 0.

3

l´ ımr→1 f (r·)−f ⋆ = 0. Note that l´ ımt→0+ f (eit·)−f ⋆ = 0 means f ∈ [eitz,BMO]. Note that l´ ımr→1 f (r·)−f ⋆ = 0 can be written l´ ımt→0+ f (e−t·)−f ⋆ = 0 Problems: 1.- Describe (ϕt) such that VMOA = [ϕt,BMOA].

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The problem for BMOA Here it is our starting motivation: Theorem A. (Sarason) Suppose f ∈ BMOA, then the following are equivalent:

1

f ∈ VMOA.

2

l´ ımt→0+ f (eit·)−f ⋆ = 0.

3

l´ ımr→1 f (r·)−f ⋆ = 0. Note that l´ ımt→0+ f (eit·)−f ⋆ = 0 means f ∈ [eitz,BMO]. Note that l´ ımr→1 f (r·)−f ⋆ = 0 can be written l´ ımt→0+ f (e−t·)−f ⋆ = 0 Problems: 1.- Describe (ϕt) such that VMOA = [ϕt,BMOA]. 2.- Given (ϕt) calculate [ϕt,BMOA].

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The case X = Bloch Definition An analytic function f is said to belong to Bloch if f Bloch = |f (0)|+sup

z∈D

|f ′(z)|(1−|z|2) < ∞,

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The case X = Bloch Definition An analytic function f is said to belong to Bloch if f Bloch = |f (0)|+sup

z∈D

|f ′(z)|(1−|z|2) < ∞, bloch is the subspace of functions such that l´ ım

|z|→1|f ′(z)|(1−|z|2) = 0

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The case X = Bloch Definition An analytic function f is said to belong to Bloch if f Bloch = |f (0)|+sup

z∈D

|f ′(z)|(1−|z|2) < ∞, bloch is the subspace of functions such that l´ ım

|z|→1|f ′(z)|(1−|z|2) = 0

It is known that bloch is the closure of polynomials in the Bloch space and (bloch)∗∗ = Bloch

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The problem for Bloch We also start with the well known result Theorem B. (Anderson-Clunie-Pommerenke) Suppose f ∈ Bloch. Then f ∈ bloch ⇐ ⇒ l´ ım

r→1f (r·)−f Bloch = 0.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The problem for Bloch We also start with the well known result Theorem B. (Anderson-Clunie-Pommerenke) Suppose f ∈ Bloch. Then f ∈ bloch ⇐ ⇒ l´ ım

r→1f (r·)−f Bloch = 0.

Problems: 3.- Does it hold that bloch = [eitz,Bloch]?

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The problem for Bloch We also start with the well known result Theorem B. (Anderson-Clunie-Pommerenke) Suppose f ∈ Bloch. Then f ∈ bloch ⇐ ⇒ l´ ım

r→1f (r·)−f Bloch = 0.

Problems: 3.- Does it hold that bloch = [eitz,Bloch]? 4.- Describe (ϕt) such that bloch = [ϕt,Bloch].

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

The problem for Bloch We also start with the well known result Theorem B. (Anderson-Clunie-Pommerenke) Suppose f ∈ Bloch. Then f ∈ bloch ⇐ ⇒ l´ ım

r→1f (r·)−f Bloch = 0.

Problems: 3.- Does it hold that bloch = [eitz,Bloch]? 4.- Describe (ϕt) such that bloch = [ϕt,Bloch]. 5.- Given (ϕt) calculate [ϕt,Bloch].

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

A basic calculation In general VMOA [ϕt,BMOA] BMOA.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

A basic calculation In general VMOA [ϕt,BMOA] BMOA. Let ϕt(z) = e−tz +1−e−t. Then f (z) = log(

1 1−z ) ∈ [ϕt,BMOA]\VMOA.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

A basic calculation In general VMOA [ϕt,BMOA] BMOA. Let ϕt(z) = e−tz +1−e−t. Then f (z) = log(

1 1−z ) ∈ [ϕt,BMOA]\VMOA. Indeed

f (ϕt(z)) = log( 1 1−ϕt(z)) = tf (z) and therefore l´ ım

t→0f ◦ϕt −f ∗ = 0.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Results on BMOA Theorem Every semigroup (ϕt) generates a semigroup of operators on VMOA, i.e. VMOA = [ϕt,VMOA].

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Results on BMOA Theorem Every semigroup (ϕt) generates a semigroup of operators on VMOA, i.e. VMOA = [ϕt,VMOA]. Theorem Let G be the infinitesimal generator of (ϕt). Then, [ϕt,BMOA] = {f ∈ BMOA : Gf ′ ∈ BMOA}.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

More results on BMOA Theorem Let (ϕt) be a semigroup with infinitesimal generator G. Assume that for some 0 < α < 1, (1−|z|)α G(z) = O (1) as |z| → 1. (2) Then VMOA = [ϕt,BMOA].

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

More results on BMOA Theorem Let (ϕt) be a semigroup with infinitesimal generator G. Assume that for some 0 < α < 1, (1−|z|)α G(z) = O (1) as |z| → 1. (2) Then VMOA = [ϕt,BMOA]. Corollary Suppose (ϕt(z)) is a semigroup whose generator G satisfies the condition (2). Then for a function f ∈ BMOA the following are equivalent

1

f ∈ VMOA.

2

l´ ımt→0+ ||f ◦ϕt −f ⋆ = 0.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Results on Bloch Theorem Any semigroup of analytic functions (ϕt) generates a C0-semigroup in bloch, i.e. [ϕt,bloch] = bloch.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Results on Bloch Theorem Any semigroup of analytic functions (ϕt) generates a C0-semigroup in bloch, i.e. [ϕt,bloch] = bloch. Theorem There are not non-trivial semigroups of analytic functions (ϕt) generating a C0-semigroup in Bloch, i.e. if [ϕt,Bloch] = Bloch then ϕt(z) = 0.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Results on Bloch Theorem Any semigroup of analytic functions (ϕt) generates a C0-semigroup in bloch, i.e. [ϕt,bloch] = bloch. Theorem There are not non-trivial semigroups of analytic functions (ϕt) generating a C0-semigroup in Bloch, i.e. if [ϕt,Bloch] = Bloch then ϕt(z) = 0. Theorem Let G be the infinitesimal generator of (ϕt). Then, [ϕt,Bloch] = {f ∈ Bloch : Gf ′ ∈ Bloch}.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Main results on Bloch and BMOA Suppose now that X is either VMOA or bloch so that the second dual X ∗∗ is BMOA or Bloch respectively. Let (ϕt) be a semigroup on D and let (Ct) be the induced semigroup of composition operators on X ∗∗ and denote St = Ct|X.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Main results on Bloch and BMOA Suppose now that X is either VMOA or bloch so that the second dual X ∗∗ is BMOA or Bloch respectively. Let (ϕt) be a semigroup on D and let (Ct) be the induced semigroup of composition operators on X ∗∗ and denote St = Ct|X. Theorem Let (ϕt) be a semigroup and X be one of the spaces VMOA or bloch. Denote by Γ the generator of the induced composition semigroup (St) on X and let λ ∈ ρ(Γ). Then (1) [ϕt,BMOA] = VMOA if and only if R(λ,Γ) = (λI −Γ)−1 is weakly compact on VMOA. (2) [ϕt,Bloch] = bloch if and only if R(λ,Γ) is weakly compact on bloch.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

A theorem and its proof Theorem Let G be the infinitesimal generator of (ϕt). Then, {f ∈ BMOA : Gf ′ ∈ BMOA} ⊂ [ϕt,BMOA]. Proof: Let f ∈ BMOA such that m := Gf ′ ∈ BMOA. First of all, one shows that (f ◦ϕt)′(z)−f ′(z) =

t

0 (m ◦ϕs)′(z)ds; for t ≥ 0, z ∈ D.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

A theorem and its proof Theorem Let G be the infinitesimal generator of (ϕt). Then, {f ∈ BMOA : Gf ′ ∈ BMOA} ⊂ [ϕt,BMOA]. Proof: Let f ∈ BMOA such that m := Gf ′ ∈ BMOA. First of all, one shows that (f ◦ϕt)′(z)−f ′(z) =

t

0 (m ◦ϕs)′(z)ds; for t ≥ 0, z ∈ D.

Next let I be an interval in ∂D and R(I) the corresponding Carleson rectangle.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

For 0 ≤ t ≤ 1 we have

R(I)
(f ◦ϕt)′(z)−f ′(z)
2 (1−|z|2)dA(z)

=

R(I)
t

0 (m ◦ϕs)′(z)ds

2

(1−|z|2)dA(z) ≤

R(I) t

1

(m ◦ϕs)′(z)
2 ds
(1−|z|2)dA(z)

where we have applied Cauchy-Schwarz in the inside integral.

Oscar Blasco New results on semigroups of analytic functions

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References The basic definitions New results on semigroups of analytic functions A theorem with proof

Hence f ◦ϕt −f ⋆ = sup

I⊆∂D

1 |I|

R(I)
(f ◦ϕt)′(z)−f ′(z)
2 (1−|z|2)dA(z)

1

2

≤ sup

I⊆∂D

1 |I|

R(I) t

1

(m ◦ϕs)′(z)
2 ds
(1−|z|2)dA(z)

1

2

≤ sup

I⊆∂D

t

1 1 |I|

R(I)
(m ◦ϕs)′(z)
2 (1−|z|2)dA(z)
ds

1

2

≤

t

1

0 m ◦ϕs2 ⋆ ds

1

2

≤ √ t sup

s∈[0,1]

m ◦ϕs⋆ ≤ √ tCm⋆ sup

s∈[0,1]

(1−log(1−ϕs(0)) ≤ C ′√ t, where we have used m ◦ψ∗ ≤ Cm∗ log(

e 1−ψ(0)) for any ψ : D → D

analytic. Therefore f ∈ [ϕt,BMOA].

Oscar Blasco New results on semigroups of analytic functions

New results on semigroups of analytic functions

OSCAR BLASCO

2013 AHA Granada, 23 May 2013 www.uv.es/oblasco

Contents

References

Contents

References

The basic definitions

Contents

References

The basic definitions

New results on semigroups of analytic functions

Contents

References

The basic definitions

New results on semigroups of analytic functions

A theorem with proof

The papers and their authors BCDMPS Semigroups of composition operators and integral operators in spaces of analytic functions

BCDMS Semigroups of composition operators in BMOA and the extension of a theorem of Sarason

Authors: B=Oscar Blasco, M=Josep Martinez (Univ. Valencia) C= Manuel Contreras, D= Santiago Diaz-Madrigal (Univ. Sevilla) S= Aristomenis Siskakis (Univ. Thessaloniki, Grecia) P=Michael Papadimitrakis (Univ. Crete)

ϕ0 is the identity in D,

ϕt+s = ϕt ◦ϕs, for all t,s ≥ 0,

ϕt(z) → ϕ0(z) = z, as t → 0, z ∈ D.

ϕ0 is the identity in D,

ϕt+s = ϕt ◦ϕs, for all t,s ≥ 0,

ϕt(z) → ϕ0(z) = z, as t → 0, z ∈ D. Examples: φt(z) = e−tz (Dilation semigroup) φt(z) = eitz (Rotation semigroup) φt(z) = e−tz +(1−e−t)

Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of (ϕt) is the function G(z) := l´ ım

ϕt(z)−z t = ∂ϕt ∂t (z)|t=0, z ∈ D.

Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of (ϕt) is the function G(z) := l´ ım

ϕt(z)−z t = ∂ϕt ∂t (z)|t=0, z ∈ D. G(ϕt(z)) = ∂ϕt(z) ∂t = G(z)∂ϕt(z) ∂z , z ∈ D, t ≥ 0. (1)

Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of (ϕt) is the function G(z) := l´ ım

ϕt(z)−z t = ∂ϕt ∂t (z)|t=0, z ∈ D. G(ϕt(z)) = ∂ϕt(z) ∂t = G(z)∂ϕt(z) ∂z , z ∈ D, t ≥ 0. (1) G has a unique representation G(z) = (bz −1)(z −b)P(z), z ∈ D where b ∈ D ( called the Denjoy-Wolff point of the semigroup) and P ∈ H (D) with ReP(z) ≥ 0 for all z ∈ D.

Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of (ϕt) is the function G(z) := l´ ım

Semigroups of operators Each semigroup of analytic functions gives rise to a semigroup (Ct) consisting of composition operators on H (D) via composition Ct(f ) := f ◦ϕt, f ∈ H (D).

Ct(f ) ∈ X for all t ≥ 0 and for every f ∈ X l´ ımt→0+ Ct(f )−f X = 0.

Ct(f ) ∈ X for all t ≥ 0 and for every f ∈ X l´ ımt→0+ Ct(f )−f X = 0. Given a semigroup (ϕt) and a Banach space X contained in H (D) we denote by [ϕt,X] the maximal closed linear subspace of X such that (ϕt) generates a semigroups of operators on it.

Previous results on semigroups of analytic functions Theorem

Every semigroup of analytic functions generates a semigroup of

Ap (1 ≤ p < ∞) and the Dirichlet space, i.e. [ϕt,X] = X in these cases.

Previous results on semigroups of analytic functions Theorem

Every semigroup of analytic functions generates a semigroup of

Ap (1 ≤ p < ∞) and the Dirichlet space, i.e. [ϕt,X] = X in these cases.

No non-trivial semigroup generates a semigroup of operators in the space H∞ of bounded analytic functions, i.e. [ϕt,H∞] = H∞ implies Φ = 0.

Previous results on semigroups of analytic functions Theorem

Every semigroup of analytic functions generates a semigroup of

Ap (1 ≤ p < ∞) and the Dirichlet space, i.e. [ϕt,X] = X in these cases.

No non-trivial semigroup generates a semigroup of operators in the space H∞ of bounded analytic functions, i.e. [ϕt,H∞] = H∞ implies Φ = 0.

There are plenty of semigroups (but not all) which generate semigroups of operators in the disk algebra.

The case X = BMOA Definition An analytic function f is said to belong to BMOA if f 2

∗ = sup I

1 |I|

where the sup is taken over all arcs I ⊂ ∂D, R(I) is the Carleson rectangle determined by I, |I| denotes the normalized length of I and dA(z) the normalized Lebesgue measure on ∂D.

The case X = BMOA Definition An analytic function f is said to belong to BMOA if f 2

∗ = sup I

1 |I|

where the sup is taken over all arcs I ⊂ ∂D, R(I) is the Carleson rectangle determined by I, |I| denotes the normalized length of I and dA(z) the normalized Lebesgue measure on ∂D. VMOA is the subspace of functions satisfying l´ ım

|I|→0

1 |I|

It is known that VMOA is the closure of the polynomials in BMOA and that (VMOA)∗∗ = BMOA.

The problem for BMOA Here it is our starting motivation: Theorem A. (Sarason) Suppose f ∈ BMOA, then the following are equivalent:

f ∈ VMOA.

l´ ımt→0+ f (eit·)−f ⋆ = 0.

l´ ımr→1 f (r·)−f ⋆ = 0.

The problem for BMOA Here it is our starting motivation: Theorem A. (Sarason) Suppose f ∈ BMOA, then the following are equivalent:

f ∈ VMOA.

l´ ımt→0+ f (eit·)−f ⋆ = 0.

l´ ımr→1 f (r·)−f ⋆ = 0. Note that l´ ımt→0+ f (eit·)−f ⋆ = 0 means f ∈ [eitz,BMO].

The problem for BMOA Here it is our starting motivation: Theorem A. (Sarason) Suppose f ∈ BMOA, then the following are equivalent:

f ∈ VMOA.

l´ ımt→0+ f (eit·)−f ⋆ = 0.

l´ ımr→1 f (r·)−f ⋆ = 0. Note that l´ ımt→0+ f (eit·)−f ⋆ = 0 means f ∈ [eitz,BMO]. Note that l´ ımr→1 f (r·)−f ⋆ = 0 can be written l´ ımt→0+ f (e−t·)−f ⋆ = 0

The problem for BMOA Here it is our starting motivation: Theorem A. (Sarason) Suppose f ∈ BMOA, then the following are equivalent:

f ∈ VMOA.

l´ ımt→0+ f (eit·)−f ⋆ = 0.

l´ ımr→1 f (r·)−f ⋆ = 0. Note that l´ ımt→0+ f (eit·)−f ⋆ = 0 means f ∈ [eitz,BMO]. Note that l´ ımr→1 f (r·)−f ⋆ = 0 can be written l´ ımt→0+ f (e−t·)−f ⋆ = 0 Problems: 1.- Describe (ϕt) such that VMOA = [ϕt,BMOA].

The problem for BMOA Here it is our starting motivation: Theorem A. (Sarason) Suppose f ∈ BMOA, then the following are equivalent:

f ∈ VMOA.

l´ ımt→0+ f (eit·)−f ⋆ = 0.

l´ ımr→1 f (r·)−f ⋆ = 0. Note that l´ ımt→0+ f (eit·)−f ⋆ = 0 means f ∈ [eitz,BMO]. Note that l´ ımr→1 f (r·)−f ⋆ = 0 can be written l´ ımt→0+ f (e−t·)−f ⋆ = 0 Problems: 1.- Describe (ϕt) such that VMOA = [ϕt,BMOA]. 2.- Given (ϕt) calculate [ϕt,BMOA].

The case X = Bloch Definition An analytic function f is said to belong to Bloch if f Bloch = |f (0)|+sup