Essential Norm of Operators into Weighted-Type Spaces on the Unit Ball

and Applied Analysis 3 see 10 and in terms of the weights. For a weight v the associated weight ṽ is defined as follows ṽ z : ( sup {∣∣f z ∣∣ : f ∈ H∞ v ,∥∥f∥∥H∞ v ≤ 1 })−1 . 1.7 For a typical weight v the associated weight ṽ is also typical. Furthermore, for each z ∈ B there is an fz ∈ H∞ v , ‖fz‖H∞ v ≤ 1, such that fz z 1/ṽ z , and the same holds for the space H0 v. It is known that H ∞ ṽ ∼ H∞ v and H0 ṽ ∼ H0 v, that is, they are isometrically isometric 10 . We say that a weight v satisfies condition L1 if it is radial and inf k∈N v ( 1 − 2− k 1 ) v ( 1 − 2−k) > 0. L1 For radial weights v satisfying condition L1 , we have that v and ṽ are equivalent, that is, there is a C ≥ 1 such that v ≤ ṽ ≤ Cv. Recently Lusky and Taskinen 12 have shown, among other results, that H0 α is isomorphic to c0. Since the closed unit ball BH∞ v is a compact subset of H B , τ , a result of Dixmier-Ng 13 gives that the subspace of H∞ v ∗ Gv : { l ∈ H∞ v ∗ : l | BH∞ v is τ-continuous } 1.8 is a predual of H∞ v , that is, G ∞ v ∗ ∼ H∞ v . Clearly the evaluation functional at z ∈ B, defined by δz f f z , belongs to Gv . The norm of δz is denoted by ‖δz‖v. Moreover, the set {δz : z ∈ B} is a total set, that is, its linear span is norm dense in Gv . More precisely, the next isomorphism result is due to Bierstedt and Summers. Lemma 1.1 see 11 . The map f → l → 〈l, f〉 is an onto isometric isomorphism between H∞ v and Gv ∗ and the restriction map l → l|H0 v gives rise to an isometric isomorphism between Gv and H0 v ∗. Similarly to the corresponding result in the one variable 14 , one can prove the following. Lemma 1.2. Suppose β > −1 and γ > 0. Then Aβ ∗ is isomorphic toH0 γ 〈 f, g 〉 β,γ ∫ B f z g z cβ γυβ γ z dV z , f ∈ H∞ γ , g ∈ Aβ. 1.9 Moreover, under the pairing 〈f, g〉β,γ with g ∈ H0 γ and f ∈ Aβ, we also have that H0 γ ∗ is isomorphic to Aβ. For z,w ∈ B, let K β z w 1 1 − 〈w, z〉 n β 1 . 1.10 4 Abstract and Applied Analysis Then the kernel function K γ z clearly belongs to H0 γ and to A 1 β . It has the reproducing property


Introduction and Preliminaries
Characterizing the compactness of composition or weighted composition operators, their differences, and Toeplitz operators between Banach spaces of analytic functions has attracted attention of numerous authors, and there has been a great interest in the matter of calculating or estimating essential norms of operators, see, for example, 1-8 .Motivated by this line of investigations, the first two authors calculated in 3 the essential norm of any operator acting between weighted-type spaces or between Bloch spaces on the unit disk and also estimated it on the weighted-Bergman space A 1 α D .We obtain a formula for the essential norm of any operator into a weighted-type space on the unit ball in C n whose domain space belongs to a general class of Banach holomorphic function spaces, thus extending to the case of the unit ball some results in 3 .Some applications of our main results are given.
Let B be the open unit ball in the euclidian complex-vector space C n and H B the space of all holomorphic functions on B. The pseudohyperbolic distance between z, u ∈ B is denoted by ρ z, u see 9 for more details .If X is a Banach space, by B X we denote the closed unit ball in X.
Let v be a positive continuous function on B weight .The weighted-type space or Bergman space of infinite order H ∞ v B H ∞ v is defined by The little weighted-type space With the norm • H ∞ v , both are Banach spaces.The norm topology of H ∞ v is finer than the topology τ of uniform convergence on compact subsets of B. For v α z 1 − |z| 2 α , α > 0, the standard weighted-type spaces, which we denote by H ∞ α and H 0 α , are obtained.These spaces appear in the study of growth conditions of analytic functions, see, for example, 10, 11 .
The Bloch-type space where μ is a weight, is the radial derivative of f, and ∇f is the complex gradient of f.The little Bloch-type space B μ,0 B B μ,0 consists of all f ∈ B μ such that where dV z is the normalized volume measure on B and Throughout this paper we assume that all weights v are typical, that is, they are radial, nonincreasing with respect to |z| and such that lim |z| → 1 − v z 0. Many results on weighted-type spaces of analytic functions and on operators between them are given in terms of the so-called associated weights see 10 and in terms of the weights.For a weight v the associated weight v is defined as follows For a typical weight v the associated weight v is also typical.Furthermore, for each z ∈ B there is an , and the same holds for the space , that is, they are isometrically isometric 10 .We say that a weight v satisfies condition L1 if it is radial and Clearly the evaluation functional at z ∈ B, defined by δ z f f z , belongs to G ∞ v .The norm of δ z is denoted by δ z v .Moreover, the set {δ z : z ∈ B} is a total set, that is, its linear span is norm dense in G ∞ v .More precisely, the next isomorphism result is due to Bierstedt and Summers.Similarly to the corresponding result in the one variable 14 , one can prove the following.
Moreover, under the pairing f, g β,γ with g ∈ H 0 γ and f ∈ A 1 β , we also have that For z, w ∈ B, let Then the kernel function K β γ z clearly belongs to H 0 γ and to A 1 β .It has the reproducing property for every function f ∈ A 1 β see 9, Theorem 2.2 .As a direct application we get that g w K Let E be a Banach space of analytic functions on B containing the constant functions.We denote its norm by • E .With • E * we denote the norm of its Banach dual space E * .Consider the following conditions on E. C1 There are positive constants s and C such that , for every f ∈ E, and for each z ∈ B. 1.12

C2
The analytic polynomials are dense on E.
C4 The linear span of the set the supremum taken in the extended real line.
It follows from C1 that the closed unit ball B E is τ-bounded, hence it is an equicontinuous and a τ-relatively compact set.Moreover, we have the following.

Proposition 1.3. Assume that E satisfies conditions C1 and C5 . Then the contraction operators
are well defined and compact.
Proof.Each given f ∈ E may be approximated uniformly on compact subsets by the sequence S k k∈N 0 of its Taylor polynomials at 0. Therefore, S k rz → f rz uniformly on B as k → ∞.Further, S k rz k∈N 0 ⊂ E is a Cauchy sequence by C5 , so it converges to some element in E.
Hence K r f ∈ E. Moreover K r is compact.Indeed, any sequence f m m∈N ⊂ B E has a τ-convergent subsequence, say itself, to an element g ∈ H B .Therefore f m rz m∈N converges uniformly on B to g rz , and again C5 yields that K r f m m∈N is a Cauchy sequence in E that converges to K r g .
We consider another condition on E.
b By C1 every functional at z ∈ B n , δ z : E → C, is bounded, and therefore Let X and Y be Banach spaces.The essential norm of a bounded operator T is the distance in the operator norm from T to the compact operators, that is, We write A B if there is a positive constant C, not depending on properties of A and B, such that A ≤ CB.We also write A B whenever A B and B A.

The Essential Norm of Operators
The next proposition is an extension of Proposition 2.1 in 5 .

Proposition 2.1.
Assume that E satisfies conditions C1 -C6 .Then there exists a sequence L m m∈N of compact operators on E such that i for any Proof.We prove that for every 0 < t < 1 and ε > 0 there is a compact operator L : E → E such that When this is done, a standard diagonal argument by taking a sequence t n ↑ 1 and a sequence of positive numbers ε n ↓ 0 will give the result.
The operator L will be constructed as a suitable finite convex combination of the operators K r and therefore by Proposition 1.3, it will be compact.
The operators I − K r : H B , τ → H B , τ are continuous, and for each f ∈ H B , we have that I − K r f → 0 in H B , τ when r → 1.By the Banach-Steinhaus theorem for Fréchet spaces I − K r → 0 uniformly on relatively compact subsets of H B , τ as r → 1; thus in particular on B E .Hence, if we pick s 1 such that t < s 1 < 1, we have lim Since B E is an equicontinuous set, the operator δ : B → E * , z → δ z is continuous, so we can find z 0 ∈ B such that sup |z|≤s 1 δ z E * δ z 0 E * .Moreover we have the inequalities Therefore, we find an r 1 ∈ 0, 1 such that sup Since the bounded set {δ z / δ z E * : z ∈ B} is equicontinuous in E * , the weak *topology coincides on it with the coarser one of the convergence on the dense subset of the polynomials, and also with the finer one of the uniform convergence on relatively compact sets in E.Moreover, observe that for each polynomial P , since condition C3 holds and P is bounded on B. This means that lim |z| → 1 δ z / δ z E * 0 on relatively compact sets in E, in particular on K r 1 B E .Hence there is an s 2 > s 1 such that for each f ∈ B E , we have sup Therefore, sup We can continue in this way and find two strictly increasing sequences s k and r k satisfying sup Let m > 1/ε and put Then by 2.11 and the fact that δ z 0 E * ≥ δ z E * for |z| ≤ s 1 , we have that 2.4 holds since sup Now we show that 2.3 holds.Similarly to above, we have that sup Moreover, by 2.12 , if s l < |z| ≤ s l 1 and f E ≤ 1, then except possibly for k l, in which case 2.17 Hence, we get for all z ∈ B and f E ≤ 1 that In light of condition C4 , we have Thus we also have 2.3 , and the statement is proved.
Next we state a formula for the norm of any operator from E to either H ∞ v or H 0 v .We omit its proof.

Theorem 2.2. Suppose
Since many operators can be written as weighted composition operators we state the following useful result.As usual, uC ϕ f u • f • ϕ where u is an analytic function on B and ϕ is an analytic self-map of B.

2.22
and since equality is attained at we get

2.24
So we get

2.25
Now we are ready for our main result.Compare it with Theorem 2.2.

Theorem 2.5. Assume that E satisfies conditions
then the same reasoning as in the first part of 3, Theorem 3.1 shows that T e ≥ .
To prove the reverse inequality, let L m m∈N be the sequence provided by Proposition 2.1.Then

2.29
Fix 0 < t < 1 and f in the unit ball of E. Then

2.30
The sequence I −L * m m∈N ⊂ L E * is bounded because of 2.2 in Proposition 2.1.Since for each fixed z ∈ B, and using 2.1 in Proposition 2.1 lim the sequence I − L * m m∈N converges to zero on every point in the total set {δ z : z ∈ B} ⊂ E * as m → ∞.So we appeal to the Banach-Steinhaus type theorem 15, III 4.5 to conclude that I − L * m m∈N converges to zero uniformly on compact subsets of E * .In particular on the image

2.33
On the other hand, for |z| > t, and as in the proof of 3, Theorem 3.1 sup Abstract and Applied Analysis Therefore, lim sup 2 And for this, it is enough to check that l • T is weak * continuous on the closed unit ball B E * * of F E * * , a fact that follows from observing that on B F , w * the weak * -topology coincides with the Hausdorff coarser one τ, for which both T and l are also continuous on bounded sets.
Since E has the λ-metric approximation property or is a dual space, we may use Axler et al. 16 Let T |H 0 γ be the restriction to H 0 γ of T .Then we can apply Theorem 2.5 so that

2.41
Since by 12 H 0 γ is isomorphic to c 0 , we obtain that  where 1/p 1/q 1. Therefore equality 2.47 follows simply bearing in mind 2.24 .

lim |z| → 1 μ
spaces with the norm f B μ |f 0 | b μ f .For the standard weight v p , p > 0, we get the p-Bloch space B p and the little p-Bloch space B β > −1, p ≥ 1, is the set of all analytic functions on B such that

Lemma 1 . 1
see 11 .The map f → l → l, f is an onto isometric isomorphism between H ∞ v and G ∞ v * and the restriction map l → l| H 0 v gives rise to an isometric isomorphism between G ∞ v and H 0 v * .

Corollary 2 . 11 . 1 −
Theorem 2.5 to L. For b , observe that L B p 0 /C ⊆ H 0 p and also that the quotient norm on B p 0 /C coincides with b p , and therefore using the quotient and the subspace duality, B p 0 /C * * B p /C.Moreover, L is τ-τ continuous on bounded sets, because R preserves the τ-τ continuity.Then use Corollary 2.8.For a given g ∈ H B , g 0 0 and a holomorphic self-map ϕ of B, the following integral-type operator on H B is introduced in 17 H B , z ∈ B.2.46This operator has also been studied later, for example, in 6, 8, 18, 19 .For a closely related operator see also20 .In 6 Stević calculated the essential norm of the operator P g ϕ : A 2 β → B μ .Motivated by this result we calculate here the essential norm of the operator P g ϕ : A p β → B μ,0 /C, but in terms of the associated weight of μ.Assume that β > −1, p > 1, g ∈ H B , g 0 0, ϕ is a holomorphic self-map of B and g z , we have RP g ϕ gC ϕ : A p β → H 0 μ /C.From this and the fact that the operator R is an onto isometry between B μ,0 /C and H 0 μ /C it follows that P g ϕ e gC ϕ e .By Corollary 2.6, we have gC ϕ e lim sup |z| →

L1
For radial weights v satisfying condition L1 , we have that v and v are equivalent, that is, there is aC ≥ 1 such that v ≤ v ≤ Cv.Recently Lusky and Taskinen 12 have shown, among other results, that H 0 α is isomorphic to c 0 .Since the closed unit ball B H ∞ v is a compact subset of H B , τ , a result of Dixmier-Ng 13 gives that the subspace of H .35 Applying Proposition 2.1 in 2.35 and letting t → 1 we obtain T e ≤ .* , it suffices to remark that T is weak * -weak * continuous, that is, * Notice that spaces like H ∞ v , the Bloch space and BMOA satisfy the assumptions on F in the above corollary.Let β > −1, γ > 0 and assume that T :A 1 β → A 1 β is a bounded operator.If T H 0 γ ⊆ H 0 γ , where T : H ∞ γ → H ∞γ is the dual operator with respect to the duality •, • β,γ , then Let R : B p /C → H ∞ p /C be defined by R f z Rf z .Then both R and R : B are onto isometries see, e.g., 17 .Therefore, if we put L R•T , we have L e T e .Since δ z v p 1 − |z| 2 −p , and R f z R * δ z f we also have that δ z / δ z v B p 0 /C * .Now for a , apply * .Hence B p 0 /C being a subspace of B p 0 fulfills conditions C1 -C6 .Set δ z f : 1 − |z| 2 p Rf z for z ∈ B and f ∈ B p /C. Corollary 2.10.a Let T : B |z| → 1 − T * δ z B p /C * .2.44 Proof.* L *