Essential Norm of Operators into Weighted-Type Spaces on the Unit Ball
The essential norm of any operator from a general Banach space of holomorphic functions on the unit ball in into the little weighted-type space is calculated. Some applications of the formula are given.
1. 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  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 .
We obtain a formula for the essential norm of any operator into a weighted-type space on the unit ball in 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 . Some applications of our main results are given.
Let be the open unit ball in the euclidian complex-vector space and the space of all holomorphic functions on . The pseudohyperbolic distance between is denoted by (see  for more details). If is a Banach space, by we denote the closed unit ball in .
Let be a positive continuous function on (weight). The weighted-type space (or Bergman space of infinite order) is defined by The little weighted-type space consists of all such that With the norm , both are Banach spaces. The norm topology of is finer than the topology of uniform convergence on compact subsets of . For , , the standard weighted-type spaces, which we denote by and , are obtained. These spaces appear in the study of growth conditions of analytic functions, see, for example, [10, 11].
The Bloch-type space consists of all such that where is a weight, is the radial derivative of , and is the complex gradient of .
The little Bloch-type space consists of all such that Both are Banach spaces with the norm . For the standard weight , , we get the -Bloch space and the little -Bloch space .
The standard weighted-Bergman space , , , is the set of all analytic functions on such that where is the normalized volume measure on and .
A weight is radial if it satisfies for every . Throughout this paper we assume that all weights are typical, that is, they are radial, nonincreasing with respect to and such that . 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 ) and in terms of the weights. For a weight the associated weight is defined as follows For a typical weight the associated weight is also typical. Furthermore, for each there is an , , such that , and the same holds for the space . It is known that and , that is, they are isometrically isometric .
We say that a weight satisfies condition if it is radial and For radial weights satisfying condition , we have that and are equivalent, that is, there is a such that . Recently Lusky and Taskinen  have shown, among other results, that is isomorphic to .
Since the closed unit ball is a compact subset of , a result of Dixmier-Ng  gives that the subspace of is a predual of , that is, . Clearly the evaluation functional at , defined by , belongs to . The norm of is denoted by . Moreover, the set is a total set, that is, its linear span is norm dense in . More precisely, the next isomorphism result is due to Bierstedt and Summers.
Lemma 1.1 (see ). The map is an onto isometric isomorphism between and and the restriction map gives rise to an isometric isomorphism between and .
Similarly to the corresponding result in the one variable , one can prove the following.
Lemma 1.2. Suppose and . Then is isomorphic to Moreover, under the pairing with and , we also have that is isomorphic to .
For , let Then the kernel function clearly belongs to and to . It has the reproducing property for every function (see [9, Theorem 2.2]). As a direct application we get that for all and for all .
Let be a Banach space of analytic functions on containing the constant functions. We denote its norm by . With we denote the norm of its Banach dual space . Consider the following conditions on .()There are positive constants and such that ()The analytic polynomials are dense on .(), when .()The linear span of the set is dense in .()There is a such that for all , where , the supremum taken in the extended real line.
It follows from that the closed unit ball is -bounded, hence it is an equicontinuous and a -relatively compact set. Moreover, we have the following.
Proposition 1.3. Assume that satisfies conditions () and (). Then the contraction operators given by are well defined and compact.
Proof. Each given may be approximated uniformly on compact subsets by the sequence of its Taylor polynomials at 0. Therefore, uniformly on as . Further, is a Cauchy sequence by (), so it converges to some element in . Hence .
Moreover is compact. Indeed, any sequence has a -convergent subsequence, say itself, to an element . Therefore converges uniformly on to , and again yields that is a Cauchy sequence in that converges to .
We consider another condition on .()For every , we have for .
Remarks 1. (a) Note that implies that .
(b) By () every functional at , , is bounded, and therefore
(c) Spaces like , the little Bloch space, Hardy spaces and weighted-Bergman spaces fulfill conditions ()–(). For , , and for , we have that (see ).
Let and be Banach spaces. The essential norm of a bounded operator is the distance in the operator norm from to the compact operators, that is,
We write if there is a positive constant , not depending on properties of and , such that . We also write whenever and .
2. The Essential Norm of Operators
The next proposition is an extension of Proposition 2.1 in .
Proposition 2.1. Assume that satisfies conditions ()–(). Then there exists a sequence of compact operators on such that (i)for any , (ii)
Proof. We prove that for every and there is a compact operator such that
When this is done, a standard diagonal argument by taking a sequence and a sequence of positive numbers will give the result.
The operator will be constructed as a suitable finite convex combination of the operators and therefore by Proposition 1.3, it will be compact.
The operators are continuous, and for each , we have that in when . By the Banach-Steinhaus theorem for Fréchet spaces uniformly on relatively compact subsets of as ; thus in particular on . Hence, if we pick such that , we have
Since is an equicontinuous set, the operator , is continuous, so we can find such that . Moreover we have the inequalities
Therefore, we find an such that
Since the bounded set is equicontinuous in , 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 . Moreover, observe that for each polynomial , since condition holds and is bounded on .
This means that on relatively compact sets in , in particular on . Hence there is an such that for each , we have Therefore,
We can continue in this way and find two strictly increasing sequences and satisfying Let and put Then by (2.11) and the fact that for , we have that (2.4) holds since
Now we show that (2.3) holds. Similarly to above, we have that
Moreover, by (2.12), if and , then except possibly for , in which case Hence, we get for all and that In light of condition (), 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 to either or . We omit its proof.
Theorem 2.2. Suppose is a bounded operator. Then
Since many operators can be written as weighted composition operators we state the following useful result. As usual, where is an analytic function on and is an analytic self-map of .
Corollary 2.3. If is a bounded weighted composition operator, then
Example 2.4. Let , , with and , . Then, since for all and , and since equality is attained at we get So we get
Now we are ready for our main result. Compare it with Theorem 2.2.
Theorem 2.5. Assume that satisfies conditions ()–(). If is a bounded operator, then
then the same reasoning as in the first part of [3, Theorem 3.1] shows that .
To prove the reverse inequality, let be the sequence provided by Proposition 2.1. Then for all .
Next, we estimate
Fix and in the unit ball of . Then
The sequence is bounded because of (2.2) in Proposition 2.1. Since for each fixed , and using (2.1) in Proposition 2.1 the sequence converges to zero on every point in the total set as . So we appeal to the Banach-Steinhaus type theorem [15, III 4.5] to conclude that converges to zero uniformly on compact subsets of . In particular on the image of the compact set , that is, Therefore since the weight is bounded, On the other hand, for , and as in the proof of [3, Theorem 3.1] Therefore, Applying Proposition 2.1 in (2.35) and letting we obtain .
Corollary 2.6. Assume that satisfies conditions ()–(). If is a bounded weighted composition operator, then
Proof. We apply Theorem 2.5 and just use that and that .
Example 2.7. If , then from Corollary 2.6 we get the following extension of the one-dimensional result in [5, Theorem 2.2]
Corollary 2.8. Let be a Banach space of analytic functions in that is isometrically isomorphic to the bidual of a space satisfying conditions ()–(). Assume that has the -metric approximation property or is a dual space. Let be a bounded operator such that and whose restrictions to bounded subsets of are continuous. Then
Proof. To see that , it suffices to remark that is weak*-weak* continuous, that is, to check that for all . And for this, it is enough to check that is weak* continuous on the closed unit ball of , a fact that follows from observing that on the weak*-topology coincides with the Hausdorff coarser one , for which both and are also continuous on bounded sets.
Since has the -metric approximation property or is a dual space, we may use Axler et al.  to obtain that . From this and Theorem 2.5 the corollary follows.
Notice that spaces like , the Bloch space and satisfy the assumptions on in the above corollary.
Corollary 2.9. Let , and assume that is a bounded operator. If , where is the dual operator with respect to the duality , then
Proof. Note that . Let be the restriction to of . Then we can apply Theorem 2.5 so that For , and , Since by  is isomorphic to , we obtain that as wanted.
Now we apply Theorem 2.5 to bounded operators on the subspaces of -Bloch and little -Bloch spaces consisting of all such that . We denote the spaces by and correspondingly. Similar subspaces of and are denoted by and correspondingly.
Observe that for any , . Hence being a subspace of fulfills conditions ()–().
Set for and .
(a) Let be a bounded operator. Then
(b) Let be a bounded operator such that is an invariant subspace for and whose restrictions to the bounded subsets are continuous. Then
Proof. Let be defined by . Then both and are onto isometries (see, e.g., ). Therefore, if we put , we have . Since , and we also have that
Now for (a), apply Theorem 2.5 to . For (b), observe that and also that the quotient norm on coincides with , and therefore using the quotient and the subspace duality, . Moreover, is continuous on bounded sets, because preserves the continuity. Then use Corollary 2.8.
For a given , and a holomorphic self-map of , the following integral-type operator on is introduced in 
This operator has also been studied later, for example, in [6, 8, 18, 19]. For a closely related operator see also . In  Stević calculated the essential norm of the operator . Motivated by this result we calculate here the essential norm of the operator , but in terms of the associated weight of .
Corollary 2.11. Assume that , , , , is a holomorphic self-map of and is bounded. Then
Proof. Since , we have . From this and the fact that the operator is an onto isometry between and it follows that . By Corollary 2.6, we have where . Therefore equality (2.47) follows simply bearing in mind (2.24).
P. Galindo is supported partially by Project MEC-FEDER 2007 064521. M. Lindström is supported partially by Magnus Ehrnrooths stiftelse and Project MEC-FEDER 2007 064521. S. Stević is supported partially by the Serbian Ministry of Education and Science (projects III44006 and III41025).
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