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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 327692, 5 pages
http://dx.doi.org/10.1155/2013/327692
Research Article

Fredholm Weighted Composition Operator on Weighted Hardy Space

School of Mathematics and Computer Science, Shanxi Normal University, Linfen 041004, China

Received 29 January 2013; Accepted 11 July 2013

Academic Editor: L. E. Persson

Copyright © 2013 Liankuo Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper gives a unified characterization of Fredholm weighted composition operator on a class of weighted Hardy spaces.

1. Introduction

Let be a Hilbert space of analytic functions on the open unit disk . For and analytic self-map of , the weighted composition operator on is defined as It is natural to ask for which and the weighted composition operator is bounded and compact on . In the past ten years, this problem on various Hilbert spaces of analytic functions has been studied extensively. It seems there is no unified ways to solve the problem since different spaces have very distinguished properties; see [110] for the characterization of bounded and compact weighted composition operators on different function spaces. In [11], the invertibility of weighted composition operator on the classical Hardy space is characterized, as an extension, in [12, 13]; the Fredholmness of weighted composition operator on Dirichlet space and weighted Dirichlet space is characterized, respectively. The results show that the Fredholmness and invertibility of weighted composition operator are closely related to properties of the reproducing kernel functions. Inspired by the characterization of Fredholm composition operator in [6], in this paper, we give a unified characterization of Fredholm weighted composition operator on a class of weighted Hardy space.

Now we follow some notations as in [14].

is called a weighted Hardy space if the monomials constitute a complete orthogonal set of nonzero vectors in . Denote ; then for with , So we use to denote the weighed Hardy space with weighted sequence .

is called automorphism invariant if, for any automorphism map of , That is, the composition operator is bounded on if is an automorphism map of . It is well known that many familiar function spaces such as Hardy space, Bergman space, and Dirichlet space are automorphism-invariant weighted Hardy space. In [14, p. 127], an example of nonautomorphism-invariant weighted Hardy space is presented.

Our main result reads as follows.

Theorem 1. Let be an automorphism-invariant weighted Hardy space such that for some nonnegative integer . If is a bounded Fredholm operator on , then there exist constants , with , such that for , and is an automorphism of  .

As an application, in the end of this paper, we give a complete characterization of Fredholm and invertible weighed composition operator on (). Recall that is a class of important weighed Hardy space with .

2. Fredholm Weighted Composition Operator on

Recall that has reproducing kernel function That is, for ,

Let then is the th derivative evaluation kernel function at in [14], that is, for ,

Firstly we give some discussion about weighted composition operator acting on . The following lemmas is well known and easy to verify.

Lemma 2. Consider .

Lemma 3. Let .

By Fań di Bruno’s formula, can be expressed as linear combination of where and with nonnegative integer .

Since it follows from Lemma 3 and (9) that can be expressed as where is algebraic combination of and with ,  .

Since the degree of nonzero coefficient is at least and the coefficient of for is so by (12) the coefficient of in for is

On the other hand, So by (10) we have

Comparing (12) and (17) with the aid of (15), we obtain

By the above reasoning, we obtain the following lemma.

Lemma 4. Consider where is linear combination of with , .

Note that in [15, Lemma 2], a different form of is given.

Now we give the characterization of Fredholm weighted composition operator on .

Proposition 5. If is a bounded Fredholm operator on , then has only finite zeroes in , and is univalent.

Proof. Let such that . By Lemma 2, which implies that if ; then . Since is a Fredholm operator, is finite; it follows that has only finite zeroes in .
If for some with , then there exists infinitely disjoint subset , of   such that . Since has only finite zeroes in , we can assume . By Lemma 2, we have which implies that , a contradiction to the Fredholmness of . So must be univalent.

Theorem 6. Let be a weighted Hardy space of such that for some nonnegative integer . If is a bounded Fredholm operator on , then is an automorphism of .

Proof. Let be the minimal nonnegative integer such that Then as . As in the proof of [14, Theorem 2.17], And for ,   is uniformly bounded for .
Since , it follows that . So
By induction and the fact that , it follows from Lemma 4 that for .
By (19), we have
As in [6, Theorem 2], we can obtain that is onto, otherwise contradicts the Fredholmness of . Combined with Proposition 5, it follows that is an automorphism of .

Theorem 6 gives the necessary condition of for the weighted composition operator to be Fredholm on with condition (22). Furthermore, if is automorphism invariant, then the multiplication operator on must be bounded Fredholm since in this case is invertible and So in the following, we consider the condition for a multiplication operator on to be Fredholm.

Denote . For , the multiplication operator on is bounded. is called the multiplier space of .

Lemma 7. Let . Then .

Proposition 8. Let be a weighted Hardy space such that for some nonnegative integer . If is a bounded Fredholm operator on , then there exist and such that for .

Proof. If is a Fredholm operator, then there exist bounded operator and compact operator such that where is identity. Acting on , we have Let ; then , since .

The following example shows that generally is not automorphism invariant even if satisfies condition (22).

Example 9. Fix . Let and for . It is easy to verify that which means that and hence , where is an automorphism of .

Now, we give the proof of the main result of Theorem 1.

Proof. By Theorem 6, is an automorphism of . Since is automorphism invariant, is bounded and invertible with . It follows that is a bounded Fredholm operator on . By Proposition 8, there exist and such that for .

3. Application

In the section, we give a complete characterization of bounded Fredholm and bounded invertible weighted composition operator on .

First, we cite some well-known results of . Denotes by the multiplier space of .(1)For , is an algebra. is the classical Dirichlet space. For ,   is equivalent to the Hilbert space with reproducing kernel function . is the classical Hardy space. For , with equivalent norms [16, 17]. (2)For ,  . For , . (For more information about , see [18].)(3)Let . is Fredholm if and only if there exist , , ,  and . is invertible if and only if there exists such that , .(4) is automorphism invariant.

By Theorem 1, (29), and the results (3), (4) above, we have the following corollaries.

Corollary 10. is a bounded Fredholm operator on if and only if , and there exist and such that for ;   is automorphism of .

Corollary 11. is a bounded invertible operator on if and only if , and there exists such that for ; is automorphism of .

Acknowledgments

This work is supported by NSFC (10971195, 11201274) and Young Science Funds of Shanxi (2010021002-2).

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