[PPT] - Volterra operators on Banach spaces of analytic functions om, PowerPoint Presentation

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Volterra operators on Banach spaces of analytic functions

Mikael Lindstr¨

m, ˚

Abo Akademi University mikael.lindstrom@abo.fi Poznan, July 2018

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 1 / 35

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Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 2 / 35

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Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 3 / 35

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Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 4 / 35

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Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 5 / 35

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Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 6 / 35

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Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 7 / 35

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Outline

1

Introduction

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 8 / 35

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Outline

1

Introduction

2

Boundedness and compactness results for Tg : H∞

vα → H∞

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 8 / 35

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Outline

1

Introduction

2

Boundedness and compactness results for Tg : H∞

vα → H∞ 3

Boundedness, weak compactness and compactness of T ϕ

g

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 8 / 35

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Introduction

The main object of study in this talk is the generalized Volterra operator T ϕ

g , defined for a fixed function g ∈ H(D) and selfmap ϕ: D → D as

T ϕ

g (f )(z) =

ϕ(z) f (ξ)g′(ξ)dξ, z ∈ D, f ∈ H(D). In the special case when ϕ is the identity map ϕ(z) = z we get the Volterra operator Tg(f )(z) = z f (ξ)g′(ξ)dξ, z ∈ D, f ∈ H(D), which has been extensively studied on various spaces of analytic functions during the past decades, starting from papers by Pommerenke, Aleman, Cima and Siskakis.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 9 / 35

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Introduction

The present talk is inspired by two recent works on boundedness and compactness properties of the Volterra operator Tg mapping into the space H∞ of bounded analytic functions on the unit disc. Namely, Smith, Stolyarov and Volberg obtained a very nice necessary and sufficient condition for Tg to be bounded on H∞ when g is univalent. The main purpose is to demonstrate that similar conditions characterize boundedness and compactness of Tg : H∞

vα → H∞ when 0 ≤ α < 1 and g

is univalent. In the other work, Contreras, Pel´ aez, Pommerenke and R¨ atty¨ a studied boundedness, compactness and weak compactness of Tg : X → H∞ acting

n a Banach space X ⊂ H(D).

We will discuss a similar study of the generalized Volterra operator T ϕ

g

mapping between Banach spaces of analytic functions on the unit disc satisfying certain general conditions.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 10 / 35

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Introduction

The weighted Banach spaces of analytic functions H∞

v

and H0

v are defined

by H∞

v

=

f ∈ H(D) : ||f ||H∞

v

:= sup

z∈ v(z)|f (z)| < +∞

H0

v =

f ∈ H∞

v

: lim

|z|→1− v(z)|f (z)| = 0

,

where the weight v : D → R is a continuous, strictly positive function, radial, non-increasing with respect to |z|, and lim|z|→1 v(z) = 0. Moreover, we will study the Bloch-type spaces Bv and B0

v defined by

Bv =

f ∈ H(D) : ||f ||Bv := |f (0)| + sup

z∈ v(z)|f ′(z)| < +∞

B0

v =

f ∈ Bv :

lim

|z|→1− v(z)|f ′(z)| = 0

.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 11 / 35

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Introduction

In 2006 Lusky has shown that H∞

v

≈ l∞ and H0

v ≈ c0 for a large class of

weights including the normal weights, i.e. weights v for which inf

k lim sup n

v(1 − 2−n−k) v(1 − 2−n) < 1 and sup

n

v(1 − 2−n) v(1 − 2−n−1) < ∞. Therefore H∞

v

and H0

v are nice spaces whenever v is a normal weight.

By the differentiation operator D : f → f ′, the spaces H∞

v

and ˜ Bv as well as their subspaces H0

v and ˜

B0

v are isometrically isomorphic. The map

(f , λ) → f + λ gives that also ˜ Bv ⊕1 C ≈ Bv and ˜ B0

v ⊕1 C ≈ B0 v are

isometrically isomorphic. Hence also Bv and B0

v behave nicely for normal weights.

Weights of the type vα(z) := (1 − |z|2)α with α > 0 are called standard weights, and they are clearly normal.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 12 / 35

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Introduction

Furthermore, the Hardy space Hp for 1 ≤ p < ∞ consists of all functions f analytic in the unit disc such that f p

Hp := sup 0≤r<1 1 2π

2π |f (reiθ)|pdθ < ∞, and the weighted Bergman spaces for constants α > −1 and 1 ≤ p < ∞ are given by Ap

α =

f ∈ H(D) : ||f ||p

Ap

α := (1 + α)

|f (z)|p(1 − |z|2)αdA(z) < ∞
,

where dA(z) is the normalized area measure on D. Finally, the disc algebra, A(D) is the space of functions analytic on D that extend continuously to the boundary ∂D.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 13 / 35

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Introduction

The following result is useful when characterizing compactness and weak compactness of the generalized Volterra operator.

Lemma (Contreras, Pel´ aez, Pommerenke, R¨ atty¨ a )

Let X ⊂ H(D) be a Banach space such that the closed unit ball BX is compact with respect to the compact open topology co and Y ⊂ H(D) be a Banach space such that point evaluation functionals on Y are bounded. Assume that T : X → Y is a co-co continuous linear operator. Then T : X → Y is compact (respectively weakly compact) if and only if {T(fn)}∞

n=1 converges to zero in the norm (respectively in the weak

topology) of Y for each bounded sequence {fn}∞

n=1 in X such that fn → 0

uniformly on compact subsets of D. The above lemma can be applied to the generalized Volterra operator T ϕ

g : X → Y, since it is obvious that T ϕ g : H(D) → H(D) always is co-co

continuous.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 14 / 35

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Boundedness and compactness results for Tg

Consider the classical Volterra operator Tg : H∞

vα → H∞ induced by a

univalent function g for standard weights vα with 0 ≤ α < 1. The study of the case when α = 0, that is when Tg : H∞ → H∞, was initiated by Anderson, Jovovic and Smith and they conjectured that T[H∞] = {g ∈ H(D) : Tg : H∞ → H∞ is bounded} would coincide with the space of functions analytic in D with bounded radial variation BRV =

f ∈ H(D) :

sup

0≤θ<2π

1 |f ′(reiθ)| dr < ∞

.

Recently Smith, Stolyarov and Volberg confirmed this conjecture when the inducing function g is univalent, that is T[H∞] ∩ {g ∈ H(D) : g is univalent} = BRV.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 15 / 35

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Boundedness and compactness results for Tg

On the other side, they also obtained a counterexample to the general conjecture posed by Anderson, Jovovic and Smith meaning that BRV T[H∞]. Contreras, Pel´ aez, Pommerenke and R¨ atty¨ a showed that Tg : H∞

v1 → H∞

is bounded precisely when g is a constant function. Since the size of the spaces H∞

vα increases as the power α grows, one concludes that the only

Volterra operator Tg : H∞

vα → H∞ that can be bounded when α ≥ 1 is the

zero operator. Hence, we are left to consider the remaining cases 0 ≤ α < 1, and will also restrict to univalent symbols g.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 16 / 35

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Boundedness and compactness results for Tg

Anderson, Jovovic and Smith also discussed compactness of the Volterra

perator Tg : H∞ → H∞, and suggested the space

BRV0 =

f ∈ H(D) : lim

t→1−

sup

0≤θ<2π

1

t

|f ′(reiθ)| dr = 0

f functions analytic in the unit disc with derivative uniformly integrable
n radii as a natural candidate for the set of such functions g.

Therefore we are interested in compactness of Tg : H∞

vα → H∞ when

0 ≤ α < 1 and g is univalent.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 17 / 35

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Boundedness and compactness results for Tg

For β > 0 and r > 0, let B

Ωr

β

denote the class of all functions F,

analytic in the open sector Ωr

β :=

z ∈ C : 0 < |z| < r and − β

2 < arg(z) < β 2

, with
F ′(z)
≤ CF

|z| for z ∈ Ωr

β. Here CF is a constant only depending on β, r and the function

F. Below Ωβ := Ω1

β and

u is the harmonic conjugate of u with u 1

2

= 0.

Theorem (Smith, Stolyarov,Volberg)

Let 0 < γ < β < π and ε > 0. Then there is a number δ(ε) > 0 such that for each F ∈ B

Ω1/2

γ

there exists a harmonic function u : Ωβ → R with

the properties (1) |Re (F(x)) − u(x)| ≤ ε, for x ∈ (0, δ(ε)]. (2) | u(z)| ≤ C(ε, γ, β, CF) < ∞, for z ∈ Ωβ. The number δ(ε) is independent of F, whereas the constant C(ε, γ, β, CF) does depend on F but only through CF as defined above.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 18 / 35

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Boundedness and compactness results for Tg

By using the above crucial result it can be proved that the following result

holds. The case α = 0 was obtained by Smith, Stolyarov and Volberg and

solved the conjecture of Anderson, Jovovic and Smith.

Theorem

If g ∈ H(D) is univalent and 0 ≤ α < 1, then Tg : H∞

vα → H∞ is bounded

if and only if sup

0≤θ<2π

1 |g′(reiθ)| (1 − r2)α dr < ∞.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 19 / 35

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Boundedness and compactness results for Tg

We now come to the compactness of Tg : H∞ → H∞, and also to the compactness of Tg : A(D) → A(D) because both statements are equivalent. The following result holds.

Theorem

If g ∈ H(D) is univalent and 0 ≤ α < 1, then Tg : H∞

vα → H∞ is compact

if and only if lim

t→1−

sup

0≤θ<2π

1

t

|g′(reiθ)| (1 − r2)α dr = 0. (1)

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 20 / 35

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Boundedness and compactness results for Tg

We now briefly indicate some steps of the proof for compactness: Assume that g does not satisfy condition (1). Then there are a constant c > 0, a sequence {tn}∞

n=1 such that 0 < tn < tn+1 < 1, limn→∞ tn = 1

and for every n ∈ N one can choose angles 0 ≤ θn < 2π such that 1

tn

|g′(reiθn)| (1 − r2)α dr > c − 1

n.

For every n ∈ N we can find a sequence of positive integers {kn}∞

n=1 such

that tkn

n ≥ 1 2 for every n ∈ N and limn→∞ kn = +∞. Now we get test

functions as follows sn(z) := fn(z)zkn and we have used the crucial result

f Smith, Stolyarov and Volberg to construct the uniformly bounded

sequence {fn}∞

n=1 of test functions in H∞ vα with certain properties. The

btained sequence {sn}∞

n=1 ⊂ H∞ vα is such that sn co

− → 0 and supn∈N snH∞

vα ≤ eC1(γ,β), where the constant C1 is independent of n.

Now Tg : H∞

vα → H∞ is non-compact, since it still holds that

lim sup

n→∞ Tg(sn)∞ ≥ 1 2 cos

π

4

e−C1(γ,β)c > 0,

and the proof is complete.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 21 / 35

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Boundedness and compactness results for Tg

Notice that it is only of interest to study boundedness and compactness of the operator Tg : H∞

vα → H∞ vβ using the approach in the proofs of the

above theorems when 0 ≤ α < 1 and β = 0, since in the other cases much better corresponding results hold. The only case for which I did not find references in the literature was when α = 0 and β > 0. However, according to a result of the next section, Tg : H∞ → H∞

vβ is bounded for β > 0 if

and only if g ∈ H∞

vβ , and the compactness can be solved as follows.

Theorem

Let g ∈ H(D) and β > 0. Then the operator Tg : H∞ → H∞

vβ is compact

if and only if g ∈ H0

vβ.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 22 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

Let X be a Banach space of analytic functions on the unit disc D that contains the constant functions, and let · X denote its norm. For any z ∈ D, the evaluation functional δz : X → C is defined as δz(f ) = f (z) forf ∈ X. We will also consider the following conditions on the space X: (I) The closed unit ball BX of X is compact with respect to the compact

pen topology co. In particular, the identity map

id: (X, · X ) → (X, co) is continuous and hence δz ∈ X ∗. (II) The evaluation functionals δz : X → C satisfy lim|z|→1 δzX→ C = ∞. (III) The linear operator Tr : X → X mapping f → fr, where fr(z) = f (rz), is compact for every 0 < r < 1. (IV) The operators Tr in (III) satisfy sup0<r<1 ||Tr||X→X < ∞. (V) The pointwise multiplication operator Mu : X → X satisfies ||Mu||X→X ||u||∞ for every u ∈ H∞, and in particular H∞ ⊂ X.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 23 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

Condition (I) holds for all the above mentioned spaces, except for H0

v , B0 v

and A(D). The spaces Hp and Ap

α satisfy all conditions (I)-(V) when

1 ≤ p < ∞ and α > −1, and the same is true for H∞

v

if the weight v is normal and equivalent to its associated weight v(z) = δz−1

H∞

v →C. But

e.g. H∞ does not satisfy (II) and in Bv condition (V) fails. Furthermore, it can be shown that δzHp = (1 − |z|2)

−1 p , δzH∞ = 1, δzAp α = (1 − |z|2) −2−α p

, δzH∞

v

= 1/˜ v(z), δzB = max{1, 1

2 log

1+|z|

1−|z|

}.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 24 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

From condition (I) we obtain by using the Dixmier-Ng theorem that the space

∗X := {ℓ ∈ X ∗ : ℓ|BX is co-continuous},

endowed with the norm induced by the dual space X ∗, is a Banach space and that the evaluation map ΦX : X → (∗X)∗, defined as the restriction ΦX (f ) = f

∗X, is an onto isometric isomorphism. In particular, ∗X is a

predual of X. Moreover, it follows from the Hahn-Banach theorem that the linear span of the set {δz : z ∈ D} is contained and norm dense in ∗X. For any weight v, the closed unit ball of H0

v is co-dense in the closed unit

ball BH∞

v , so by a result of Bierstedt-Summers, the restriction map

l → l|H0

v gives rise to an isometric isomorphism ∗H∞

v

≈ (H0

v )∗. Since for

normal weights v, we have H0

v ≈ c0, we conclude that ∗H∞ v

≈ l1 if the weight v is normal.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 25 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

The first result estimate the norm of T ϕ

g mapping into the spaces H∞ v

and B∞

v

in terms of the inducing symbols ϕ and g.

Theorem

Let X be a Banach space of analytic functions on D satisfying (I). (i) If the weight v is normal, then T ϕ

g X→H∞

v

≍ sup

z∈D

(1 − |z|)v(z)|(g ◦ ϕ)′(z)|δϕ(z)X→C. (ii) For any weight v, T ϕ

g X→B∞

v = sup

z∈D

v(z)|(g ◦ ϕ)′(z)|δϕ(z)X→C.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 26 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

We now turn to investigate weak compactness and compactness of the generalized Volterra operator. The following lemma ensures the existence

f a predual operator under very general assumptions.

Lemma

Let X, Y ⊂ H(D) be Banach spaces satisfying condition (I). If the

perator T : X → Y is bounded and the restriction T|BX is co-co

continuous, then the operator ΦY ◦ T ◦ Φ−1

X : (∗X)∗ → (∗Y)∗

is σ((∗X)∗, ∗X) − σ((∗Y)∗, ∗Y) continuous, and consequently there exists a bounded operator S : ∗Y → ∗X such that T = Φ−1

Y ◦ S∗ ◦ ΦX .

Using this predual operator S one can show that compactness and weak compactness coincide for a large class of operators mapping from H∞

v

r

B∞

v

into a Banach space Y ⊂ H(D).

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 27 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

Theorem

Let v be a normal weight and assume that the Banach space Y ⊂ H(D) satisfies (I). (i) If the restriction T|BH∞

v

is co-co continuous then T : H∞

v

→ Y is compact if and only if it is weakly compact. (ii) If the restriction T|BB∞

v

is co-co continuous then T : B∞

v → Y is

compact if and only if it is weakly compact.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 28 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

Proof.

To prove (i), suppose T : H∞

v

→ Y is weakly compact. By the above lemma there exists a bounded operator S : ∗Y → ∗H∞

v

such that T = Φ−1

Y ◦ S∗ ◦ ΦH∞

v , and hence by the Gantmacher theorem

S : ∗Y → ∗H∞

v

is weakly compact. Moreover, ∗H∞

v

is isomorphic to ℓ1 since v is normal. Thus T = Φ−1

Y ◦ S∗ ◦ ΦH∞

v

is compact.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 29 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

Since every bounded operator T : ℓ∞ → X mapping into a Banach space X not containing a copy of ℓ∞ is weakly compact, we obtain the following result.

Corollary

Let Y ⊂ H(D) be a Banach space satisfying condition (I) and not fixing a copy of ℓ∞. If v is a normal weight and T : H∞

v

→ Y is a co-co continuous linear operator, then T : H∞

v

→ Y is bounded if and only if T : H∞

v

→ Y is compact. The same statement holds when H∞

v

is replaced with the Bloch space B∞

v .

The essential norm of a bounded linear operator T : X → Y is defined to be the distance to the compact operators, that is Te,X→Y = inf

T − KX→Y : K : X → Y is compact
.

Notice that T : X → Y is compact if and only if Te,X→Y = 0.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 30 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

The next theorem gives a nice estimate of the essential norm of T ϕ

g : X → H∞ v . As we have seen earlier the situation changes dramatically

if the target space is instead H∞.

Theorem

Let X ⊂ H(D) be a Banach space satisfying conditions (I)-(V). (i) If the weight v is normal, then T ϕ

g e,X→H∞

v

≍ lim sup

|ϕ(z)|→1

(1 − |z|)v(z)|(g ◦ ϕ)′(z)|δϕ(z)X→ C. (ii) For any weight v, T ϕ

g e,X→B∞

v ≍ lim sup

|ϕ(z)|→1

v(z)|(g ◦ ϕ)′(z)|δϕ(z)X→ C.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 31 / 35

SLIDE 34

Boundedness, weak compactness and compactness of T ϕ

g

The previous essential norm estimates can be applied to the Hardy and weighted Bergman spaces.

Corollary

Let α > −1 and 1 ≤ p < ∞. If the weight v is normal, then (i) T ϕ

g e,Hp→H∞

v

≍ lim sup|ϕ(z)|→1

1−|z| (1−|ϕ(z)|)1/p v(z)|(g ◦ ϕ)′(z)|.

(ii) T ϕ

g e,Ap

α→H∞ v

≍ lim sup|ϕ(z)|→1

1−|z| (1−|ϕ(z)|)(2+α)/p v(z)|(g ◦ ϕ)′(z)|.

Also, for any weight v (iii) T ϕ

g e,Hp→B∞

v ≍ lim sup|ϕ(z)|→1

v(z) (1−|ϕ(z)|)1/p |(g ◦ ϕ)′(z)|.

(iv) T ϕ

g e,Ap

α→B∞ v ≍ lim sup|ϕ(z)|→1

v(z) (1−|ϕ(z)|)(2+α)/p |(g ◦ ϕ)′(z)|.

Example 1. Let 1 ≤ p < ∞ and p − 2 < α < ∞. Then Tg : Ap

α → H∞

2+α p −1

is compact if and only if lim|z|→1 |g′(z)| = 0, that is g is a constant. HenceTg : Ap

α → H∞

2+α p −1 = B∞ 2+α p

is compact if and only if Tg = 0.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 32 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

Next we will relate compactness of T ϕ

g : X → H∞ v

to compactness of T ϕ

g : X → H0 v , and also includes a corresponding comparison for the target

spaces B∞

v

and B0

v.

Theorem

Let X ⊂ H(D) be a Banach space. (i) If the space X satisfies conditions (I)-(V) and the weight v is normal, then T ϕ

g : X → H∞ v

is compact if and only if T ϕ

g : X → H0 v is

compact. (ii) If the space X satisfies conditions (I) and (IV), then for any weight v, T ϕ

g : X → B0 v is compact if and only if T ϕ g : X → B∞ v

is compact and g ◦ ϕ ∈ B0

v.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 33 / 35

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Boundedness, weak compactness and compactness of T ϕ

g

In general for normal weights v and w, compactness of T ϕ

g : B∞ v → B∞ w

does not imply that g ◦ ϕ ∈ B0

w.

Example 2. For each 0 < α < 1, the Volterra operator Tg : B∞

vα → B∞ vα is

compact if and only if the multiplication operator Mg′ : B∞

vα → H∞ vα = B∞ vα+1 is compact, and this happens precisely when

g ∈ B0

v1. Let 0 < α < 1 and 0 < δ < 1 − α and define gδ(z) := (1 − z)δ

for z ∈ D. Then it follows that gδ ∈ B0

v1 and therefore Tgδ : B∞ vα → B∞ vα is

compact. For each r ∈ (0, 1) we have

(1 − r2)α|g′

δ(r)| = δ(1 − r)α(1 + r)α

(1 − r)1−δ ≥ δ(1 − r)α+δ−1, which implies that gδ / ∈ B0

vα.

Mikael Lindstr¨

m ()

Volterra operators Poznan, July 2018 34 / 35

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