`Abstract and Applied AnalysisVolumeΒ 2008, Article IDΒ 196498, 12 pageshttp://dx.doi.org/10.1155/2008/196498`
Research Article

## Compact Weighted Composition Operators and Multiplication Operators between Hardy Spaces

1Department of Mathematics, Faculty of Science Division II, Tokyo University of Science, 4-6-1 Higashicho, Hitachi, Ibaraki 317-0061, Japan
2Department of Mathematics, University of Science and Technology of China, Hefei 230026, China

Received 27 August 2007; Accepted 10 February 2008

Copyright Β© 2008 Sei-Ichiro Ueki and Luo Luo. 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

We estimate the essential norm of a compact weighted composition operator acting between different Hardy spaces of the unit ball in . Also we will discuss a compact multiplication operator between Hardy spaces.

#### 1. Introduction

Let be a fixed integer. Let denote the unit ball of and let denote the space of all holomorphic functions in . For each , the Hardy spaceββ is defined bywhere is the normalized Lebesgue measure on the boundary of .

For a given holomorphic self-map of and holomorphic function in , the weighted composition operator is defined by . In particular, if is the constant function , then becomes the composition operator . In the special case that is the identity mapping of , is called the multiplication operator and is denoted by .

Let and be Banach spaces. For a bounded linear operator , the essential norm is defined to be the distance from to the set of the compact operators , namely, where denotes the usual operator norm. Clearly, is compact if and only if . Thus, the essential norm is closely related to the compactness problem of concrete operators. Many mathematicians have studied the essential norm of various concrete operators. For these studies about composition operators on Hardy spaces of the unit disc, refer to [1β4]. In this paper, our objects are weighted composition operators between Hardy spaces of the unit ball . Several authors have also studied weighted composition operators on various analytic function spaces. For more information about weighted composition operators, refer to [5β10].

Recently, Contreras and HernΓ‘ndez-DΓ­az [11, 12] have characterized the compactness of from into in terms of the pull-back measure. Here, denotes the open unit disc in the complex plane. But they have not given the estimate for the essential norm of . The essential norm of has been studied by ukoviΔ and Zhao [13, 14]. In the higher-dimensional case, Ueki [15] characterized the boundedness and compactness of , in terms of the pull-back measure and the integral operator. The purpose of this paper is to estimate the essential norm of . The following theorem is our main result.

Main 1 Theorem. Let . If is a bounded weighted composition operator from into , thenwhere is the pull-back measure induced by and , is the Carleson set of , and the notation means that the ratios of two terms are bounded below and above by constants dependent upon the dimension and other parameters.

The one variable case of the first estimate for in above theorem may be found in the work [14] by ukoviΔ and Zhao. In the case and , Choe [1] and Luo [16] showed that the essential norm is comparable to the value .

We give the proof of main theorem in Section 3. The ideas of our proofs are based on the method which Choe or Luo used in their papers. In Section 4, we have a discussion on the compact multiplication operator between different Hardy spaces.

Throughout the paper, the symbol denotes a positive constant, possibly different at each occurrence, but always independent of the function and other parameters or .

#### 2. Carleson-type Measures

For each , we can define a finite positive Borel measure on bywhere denotes the radial limit map of the mapping considered as a map of . A change-of-variable formula from measure theory shows thatfor each nonnegative measurable function on . This type of pull-back measure played an important role in past studies of composition operators on Hardy spaces of .

For each and , letIt is well known that is comparable to ([17, page 67]).

The proof of the following lemma is essentially the same as that of Power's theorem in [18].

Lemma 2.1. Let . Suppose that is a positive Borel measure on and that there exists a constant such that Then there exists a constant such that

Proof. Fix and . By the same argument as in the proof of theorem in [18, pages 14-15], it follows from (2.4) that there exists a constant such thatwhere is the admissible maximal function of which is defined byfor . By (2.6), we haveSince , it follows from [17, Theorem 5.6.5] thatBy (2.8) and (2.9), we haveThis completes the proof.

Lemma 2.2. Let . Suppose that is a positive Borel measure on such that for some constant .
(a) If , then there exist a and a constant ( is the product of and a constant depending only on the dimension ) such that and .(b) If , then for all Borel sets of .

Proof. Part (a) is completely analogous to [19, page 238, Lemma 1.3]. So we only prove part (b). Combining with (2.11), we havefor all and . Hence we see that the maximal function of the positive measure satisfies for all . By [17, page 70, Theorem 5.2.7], we obtain for some . By (2.12), we havefor all and . Letting , we see that a.e. on , and so . This completes the proof of part (b).

Combining Lemma 2.1 with Lemma 2.2 and using the same argument as in [19, page 239], we obtain the following lemma.

Lemma 2.3. Let . Suppose that is a positive Borel measure on such that for some constant . Then, there exists a constant such that for all . Here, the notation denotes the function defined on by in and a.e. on .

Remark 2.4. In Lemma 2.3 (or in Lemma 2.1), we see that the constant of (2.15) (or (2.5)) can be chosen to be the product of and a positive constant depending only on , and the dimension .

In order to prove the main theorem, we need some results. These are the extensions of [19, Corollary 1.4 and Lemma 1.6] to the case of weighted composition operators .

Proposition 2.5. Let . Suppose that is a holomorphic map and such that is bounded. Then cannot carry a set of positive -measure in into a set of -measure in .

Proof. Suppose that and with and . Put . As in the case of composition operators, it is well known that the boundedness of impliesfor some positive constant (see [15]). By Lemma 2.2, we see that (if ) or is absolutely continuous with respect to (if ). Thus we haveThat is, a.e. on . Hence [17, page 83, Theorem 5.5.9] gives that in . This contradicts .

Lemma 2.6. Let and . Suppose that is a holomorphic map and such that is bounded. Then a.e. on . Here the notation is used as in Lemma 2.3.

Proof. Since cannot carry a set of positive measure in into a set of measure in (by Proposition 2.5) and since the radial limit of , and exist on a set of full measure in , we have a.e. on .
On the other hand, since is in the ball algebra and as in , the boundedness of shows thatThis implies that a.e. on .

#### 3. Weighted Composition Operators between Hardy Spaces

Theorem 3.1. Let . If is a bounded weighted composition operator from into , then

Proof of the lower estimates. For each , we define the function on byThen the functions belong to the ball algebra and form a bounded sequence of . Take a compact operator arbitrarily. Since the bounded sequence converges to uniformly on compact subsets of as , we have as . Thus we obtainBy the definition of , we also see thatCombining this with (3.3), we getSince this holds for every compact operator , it follows that
Furthermore, we put for each and in the definition of . Since we see that for all , we haveLetting , we getCombining this with (3.6), we obtaincompleting the proof of the lower estimates.

To prove the upper estimates, we need some technical results about the polynomial approximation of . Recall that a holomorphic function in has the homogeneous expansionwhere is a multi-index, , and . For the homogeneous expansion of and any integer , letand , where is the identity operator.

Proposition 3.2. Suppose that is a Banach space of holomorphic functions in with the property that the polynomials are dense in . Then as if and only if .

Proof. We see that [20, Proposition 1] also holds if we replace the unit disc with the unit ball. So we omit the proof of this proposition.

Proposition 3.3. If , then as for each .

Proof. For each and , the slice function of is in the disc algebra . Here, denotes the dilated function of , that is . Hence [20, Corollary 3 and Proposition 1] implies that there is a positive constant such thatfor every integer . Since , integration by slices (see [17, page 15, Proposition 1.4.7.]) showsthat is, for every integer . By Proposition 3.2, we see as . This completes the proof of the proposition.

Corollary 3.4. If , then converges to pointwise in as . Moreover, .

Proof. Since , Proposition 3.3 shows that as . Furthermore, the principle of uniform boundedness implies that .

Lemma 3.5. Let . For each and ,

Proof. Let be the reproducing kernel for and let be the Cauchy-SzegΓΆ projection. Then, the orthogonality of monomials implies thatHΓΆlder's inequality and the expansion of giveThis completes the proof.

The following lemma is well known in the case of functional Hilbert spaces (cf. [4, 21]). As in the proof of [21, Lemma 3.16], an elementary argument verifies Lemma 3.6.

Lemma 3.6. Let . If is bounded from into , then

Let us prove the upper estimates for the essential norm of .

Proof of the upper estimates. For the sake of convenience, we set
By the notation (3.18), for given , we can choose an such thatfor with . For each and , we put . Since the function satisfies for all , the inequality (3.21) implies thatfor all and all .
By the notation (3.19), we can also choose a , so thatfor all . Let and be the restrictions of to and , respectively. We claim that also satisfies the Carleson measure conditionfor all and . By (3.22) or (3.23), these conditions are true for all . Hence, we assume that . For a finite cover , where of the set , the covering property implies that there exists a disjoint subcollection of so thatFurthermore, we obtain . By the notation (3.20), we havewhere the constant depends only on , and the dimension .
Now, we take a function with . By Lemma 2.6, we havefor all integers . Condition (3.24) and Lemma 2.3 implies thatOn the other hand, by Lemma 3.5, we haveThe boundedness of implies that and the convergence of the series implies thatSo we obtainCombining (3.27), (3.28), and (3.31) with Lemma 3.6, we haveSince Corollary 3.4 implies that , and was arbitrary, we conclude thatwhich were to be proved.

Corollary 3.7 (see [15]). Suppose that . For the bounded weighted composition operator , the following conditions are equivalent:
(a) is compact;(b) and satisfy(c) and satisfy

#### 4. Multiplication Operators between Hardy Spaces

In this section, we consider the compact multiplication operator between Hardy spaces. As a consequence of Theorem 3.1, we obtain the following results.

Corollary 4.1. Suppose that . For the bounded multiplication operator , the following inequality holds: Furthermore, is compact if and only if

By using Corollary 4.1, we can completely characterize the compactness of a multiplication operator from into .

Theorem 4.2. Suppose that . Then is compact if and only if in .

Proof. If , then is compact. Thus, we only prove that the compactness of implies . The boundedness of implies that . Hence, the Poisson representation for gives thatwhere is the Poisson kernel. HΓΆlder's inequality shows thatwhere . By the assumption , we see thatand so we haveInequality (4.4) and Corollary 4.1 give that . Since , this implies that has a -limit on a set of positive -measure in . Hence [17, page 83, Theorem 5.5.9] shows that . This completes the proof.

#### Acknowledgment

The authors would like to thank the referee for the careful reading of the first version of this paper and for the several suggestions made for improvement.

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