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Abstract and Applied Analysis
VolumeΒ 2008, Article IDΒ 196498, 12 pages
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

Academic Editor: StephenΒ Clark

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.


We estimate the essential norm of a compact weighted composition operator π‘’πΆπœ‘ acting between different Hardy spaces of the unit ball in ℂ𝑁. Also we will discuss a compact multiplication operator between Hardy spaces.

1. Introduction

Let 𝑁 be a fixed integer. Let 𝐡𝑁 denote the unit ball of ℂ𝑁 and let 𝐻(𝐡𝑁) denote the space of all holomorphic functions in 𝐡𝑁. For each 𝑝,1≀𝑝<∞, the Hardy space   𝐻𝑝(𝐡𝑁) is defined by𝐻𝑝𝐡𝑁𝐡={π‘“βˆˆπ»π‘ξ‚βˆΆsup0<π‘Ÿ<1βˆ«πœ•π΅π‘||||||𝑓(π‘Ÿπœ)π‘β€–β€–π‘“β€–β€–π‘‘πœŽ(𝜁)<∞},𝑝=ξ‚ƒβˆ«πœ•π΅π‘|||π‘“βˆ—|||(𝜁)π‘ξ‚„π‘‘πœŽ(𝜁)1/𝑝,(1.1)where π‘‘πœŽ is the normalized Lebesgue measure on the boundary πœ•π΅π‘ of 𝐡𝑁.

For a given holomorphic self-map πœ‘ of 𝐡𝑁 and holomorphic function 𝑒 in 𝐡𝑁, the weighted composition operator π‘’πΆπœ‘ is defined by π‘’πΆπœ‘π‘“=𝑒(𝑓0π‘₯025𝑒6πœ‘). In particular, if 𝑒 is the constant function 1, then π‘’πΆπœ‘ becomes the composition operator πΆπœ‘. In the special case that πœ‘ is the identity mapping of 𝐡𝑁, π‘’πΆπœ‘ is called the multiplication operator and is denoted by 𝑀𝑒.

Let 𝑋 and π‘Œ be Banach spaces. For a bounded linear operator π‘‡βˆΆπ‘‹β†’π‘Œ, the essential norm ‖𝑇‖𝑒,π‘‹β†’π‘Œ is defined to be the distance from 𝑇 to the set of the compact operators 𝒦, namely,‖‖𝑇‖‖𝑒,π‘‹β†’π‘Œβ€–β€–β€–β€–=inf{π‘‡βˆ’π’¦βˆΆπ’¦iscompactfrom𝑋intoπ‘Œ},(1.2) where β€–β‹…β€– denotes the usual operator norm. Clearly, 𝑇 is compact if and only if ‖𝑇‖𝑒,π‘‹β†’π‘Œ=0. 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 [1–4]. In this paper, our objects are weighted composition operators between Hardy spaces of the unit ball 𝐡𝑁. Several authors have also studied weighted composition operators on various analytic function spaces. For more information about weighted composition operators, refer to [5–10].

Recently, Contreras and HernΓ‘ndez-DΓ­az [11, 12] have characterized the compactness of π‘’πΆπœ‘ from 𝐻𝑝(𝐡1) into π»π‘ž(𝐡1)(1<π‘β‰€π‘ž<∞) in terms of the pull-back measure. Here, 𝐡1 denotes the open unit disc in the complex plane. But they have not given the estimate for the essential norm of π‘’πΆπœ‘. The essential norm of π‘’πΆπœ‘βˆΆπ»π‘(𝐡1)β†’π»π‘ž(𝐡1) has been studied by ⌣Cu⌣ckoviΔ‡ and Zhao [13, 14]. In the higher-dimensional case, Ueki [15] characterized the boundedness and compactness of π‘’πΆπœ‘βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁), in terms of the pull-back measure and the integral operator. The purpose of this paper is to estimate the essential norm of π‘’πΆπœ‘βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁). The following theorem is our main result.

Main 1 Theorem. Let 1<π‘β‰€π‘ž<∞. If π‘’πΆπœ‘is a bounded weighted composition operator from 𝐻𝑝(𝐡𝑁)into π»π‘ž(𝐡𝑁), thenβ€–β€–π‘’πΆπœ‘β€–β€–π‘žπ‘’,π»π‘β†’π»π‘ž||∼limsup𝑀||β†’1βˆ’ξ€œπœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2|||ξ‚¬πœ‘1βˆ’βˆ—ξ‚­|||(𝜁),𝑀2}π‘žπ‘/π‘π‘‘πœŽ(𝜁)∼limsup𝑑→0+supπœβˆˆπœ•π΅π‘πœ‡πœ‘,𝑒𝑆(𝜁,𝑑)π‘‘π‘žπ‘/𝑝,(1.3)where πœ‡πœ‘,𝑒 is the pull-back measure induced by πœ‘ and 𝑒, 𝑆(𝜁,𝑑) is the Carleson set of 𝐡𝑁, and the notation ∼ means that the ratios of two terms are bounded below and above by constants dependent upon the dimension 𝑁 and other parameters.

The one variable case of the first estimate for β€–π‘’πΆπœ‘β€–π‘’ in above theorem may be found in the work [14] by ⌣Cu⌣ckoviΔ‡ and Zhao. In the case 𝑝=π‘ž=2 and 𝑒=1, Choe [1] and Luo [16] showed that the essential norm β€–πΆπœ‘β€–π‘’ is comparable to the value limsup𝑑→0+supπœβˆˆπœ•π΅π‘(πœ‡πœ‘(𝑆(𝜁,𝑑))/𝑑𝑁).

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 𝐢 denotes a positive constant, possibly different at each occurrence, but always independent of the function 𝑓 and other parameters π‘Ÿ or 𝑑.

2. Carleson-type Measures

For each π‘’βˆˆπ»π‘ž(𝐡𝑁), we can define a finite positive Borel measure πœ‡πœ‘,𝑒 on 𝐡𝑁 byπœ‡πœ‘,π‘’ξ€œ(𝐸)=πœ‘βˆ—βˆ’1(𝐸)|||π‘’βˆ—|||π‘žπ‘‘πœŽ(βˆ€Borelsets𝐸of𝐡𝑁),(2.1)where πœ‘βˆ— denotes the radial limit map of the mapping πœ‘ considered as a map of πœ•π΅π‘β†’π΅π‘. A change-of-variable formula from measure theory shows thatξ€œπ΅π‘π‘”π‘‘πœ‡πœ‘,𝑒=ξ€œπœ•π΅π‘|||π‘’βˆ—|||π‘žξ‚€π‘”0π‘₯025𝑒6πœ‘βˆ—ξ‚π‘‘πœŽ,(2.2)for each nonnegative measurable function 𝑔 on 𝐡𝑁. This type of pull-back measure played an important role in past studies of composition operators on Hardy spaces of 𝐡𝑁.

For each πœβˆˆπœ•π΅π‘ and 𝑑>0, let𝑆(𝜁,𝑑)={π‘§βˆˆπ΅π‘βˆΆ||||||1βˆ’βŸ¨π‘§,𝜁⟩<𝑑},𝐡(𝜁,𝑑)=𝑆(𝜁,𝑑)βˆ©π΅π‘,𝑄(𝜁,𝑑)=𝑆(𝜁,𝑑)βˆ©πœ•π΅π‘.(2.3)It is well known that 𝜎(𝑄(𝜁,𝑑)) is comparable to 𝑑𝑁 ([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 1≀𝛼<∞. Suppose that πœ‡ is a positive Borel measure on 𝐡𝑁 and that there exists a constant 𝐢>0 such that πœ‡ξ‚€ξ‚π΅(𝜁,𝑑)β‰€πΆπ‘‘π›Όπ‘ξ‚€πœβˆˆπœ•π΅π‘ξ‚,𝑑>0.(2.4)Then there exists a constant 𝐾>0 such that ξ‚ƒξ€œπ΅π‘||𝑓||π‘π›Όξ‚„π‘‘πœ‡1/π‘π›Όβ€–β€–π‘“β€–β€–β‰€πΎπ»π‘ξ‚€π‘“βˆˆπ»π‘ξ‚€π΅π‘ξ‚ξ‚.(2.5)

Proof. Fix π‘“βˆˆπ»π‘(𝐡𝑁) and 𝑠>0. 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 𝐢>0 such thatπœ‡ξ‚€{π‘§βˆˆπ΅π‘βˆΆ||||||ξ‚ξ‚ƒπœŽξ‚€π‘“(𝑧)β‰₯𝑠}≀𝐢{πœβˆˆπœ•π΅π‘βˆΆπ‘€π‘“(𝜁)β‰₯𝑠}𝛼,(2.6)where 𝑀𝑓 is the admissible maximal function of 𝑓 which is defined by||||||𝑀𝑓(𝜁)=sup{𝑓(𝑧)βˆΆπ‘§βˆˆβ„‚π‘›,||||||||𝑧||1βˆ’βŸ¨π‘§,𝜁⟩<1βˆ’2},(2.7)for πœβˆˆπœ•π΅π‘. By (2.6), we haveξ€œπ΅π‘||𝑓||π‘π›Όξ€œπ‘‘πœ‡=π‘π›Όβˆž0||𝑓||πœ‡{>𝑠}π‘ π‘π›Όβˆ’1ξ€œπ‘‘π‘ β‰€πΆπ‘π›Όβˆž0𝜎{𝑀𝑓β‰₯𝑠}π›Όπ‘ π‘π›Όβˆ’1𝑑𝑠.(2.8)Since π‘“βˆˆπ»π‘(𝐡𝑁), it follows from [17, Theorem 5.6.5] that𝜎{𝑀𝑓β‰₯𝑠}π›Όβˆ’1π‘ π‘π›Όβˆ’π‘β‰€ξ‚ƒξ€œπœ•π΅π‘{𝑀𝑓(𝜁)}π‘ξ‚„π‘‘πœŽ(𝜁)π›Όβˆ’1‖‖𝑓‖‖≀𝐢𝐻𝑝(π›Όβˆ’1)𝑝.(2.9)By (2.8) and (2.9), we haveξ€œπ΅π‘||𝑓||π‘π›Όβ€–β€–π‘“β€–β€–π‘‘πœ‡β‰€πΆπ»π‘(π›Όβˆ’1)π‘π‘ξ€œβˆž0𝜎{𝑀𝑓β‰₯𝑠}π‘ π‘βˆ’1‖‖𝑓‖‖𝑑𝑠≀𝐢𝐻𝑝(π›Όβˆ’1)π‘ξ€œπœ•π΅π‘{𝑀𝑓(𝜁)}π‘β€–β€–π‘“β€–β€–π‘‘πœŽ(𝜁)≀𝐢𝐻𝑝𝛼𝑝.(2.10)This completes the proof.

Lemma 2.2. Let 1≀𝛼<∞. Suppose that πœ‡ is a positive Borel measure on πœ•π΅π‘ such that πœ‡ξ‚€ξ‚π‘„(𝜁,𝑑)β‰€πΆπ‘‘π›Όπ‘ξ‚€πœβˆˆπœ•π΅π‘ξ‚,𝑑>0,(2.11)for some constant 𝐢>0.
(a) If 𝛼=1, then there exist a π‘”βˆˆπΏβˆž(πœ•π΅π‘) and a constant 𝐢′>0 (𝐢′ is the product of 𝐢 and a constant depending only on the dimension 𝑁) such that π‘‘πœ‡=π‘”π‘‘πœŽ and β€–π‘”β€–πΏβˆžβ‰€πΆβ€².(b) If 𝛼>1, then πœ‡β‰‘0 for all Borel sets of πœ•π΅π‘.

Proof. Part (a) is completely analogous to [19, page 238, Lemma 1.3]. So we only prove part (b). Combining 𝜎(𝑄(𝜁,𝑑))βˆΌπ‘‘π‘ with (2.11), we haveπœ‡ξ‚€ξ‚π‘„(𝜁,𝑑)πœŽξ‚€ξ‚π‘„(𝜁,𝑑)≀𝐢𝑑𝑁(π›Όβˆ’1)(2.12)for all πœβˆˆπœ•π΅π‘ and 𝑑>0. Hence we see that the maximal function π‘€πœ‡ of the positive measure πœ‡ satisfies π‘€πœ‡(𝜁)<∞ for all πœβˆˆπœ•π΅π‘. By [17, page 70, Theorem 5.2.7], we obtain π‘‘πœ‡=π‘”π‘‘πœŽ for some π‘”βˆˆπΏ1(πœ•π΅π‘). By (2.12), we have10β‰€πœŽξ‚€ξ‚ξ€œπ‘„(𝜁,𝑑)𝑄(𝜁,𝑑)πœ‡ξ‚€ξ‚π‘”π‘‘πœŽ=𝑄(𝜁,𝑑)πœŽξ‚€ξ‚π‘„(𝜁,𝑑)≀𝐢𝑑𝑁(π›Όβˆ’1)(2.13)for all πœβˆˆπœ•π΅π‘ and 𝑑>0. Letting 𝑑→0+, we see that 𝑔=0 a.e. on πœ•π΅π‘, and so πœ‡β‰‘0. 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 1<π‘β‰€π‘ž<∞. Suppose that πœ‡ is a positive Borel measure on 𝐡𝑁 such that πœ‡ξ‚€ξ‚π‘†(𝜁,𝑑)β‰€πΆπ‘‘π‘žπ‘/π‘ξ‚€πœβˆˆπœ•π΅π‘ξ‚,𝑑>0,(2.14)for some constant 𝐢>0. Then, there exists a constant 𝐾>0 such that ξ‚ƒξ€œπ΅π‘|||π‘“βˆ—|||π‘žξ‚„π‘‘πœ‡1/π‘žβ‰€πΎβ€–β€–π‘“β€–β€–π»π‘,(2.15)for all π‘“βˆˆπ»π‘(𝐡𝑁). Here, the notation π‘“βˆ— denotes the function defined on 𝐡𝑁 by π‘“βˆ—=𝑓 in 𝐡𝑁 and π‘“βˆ—=limπ‘Ÿβ†’1βˆ’π‘“π‘Ÿ a.e. [𝜎] on πœ•π΅π‘.

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

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 π‘’πΆπœ‘.

Proposition 2.5. Let 1<π‘β‰€π‘ž<∞. Suppose that πœ‘βˆΆπ΅π‘β†’π΅π‘ is a holomorphic map and π‘’βˆˆπ»π‘ž(𝐡𝑁)⧡{0} such that π‘’πΆπœ‘βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁) is bounded. Then πœ‘βˆ— cannot carry a set of positive 𝜎-measure in πœ•π΅π‘ into a set of 𝜎-measure 0 in πœ•π΅π‘.

Proof. Suppose that 𝐸,πΉβŠ‚πœ•π΅π‘ and πœ‘βˆ—(𝐸)βŠ‚πΉ with 𝜎(𝐸)>0 and 𝜎(𝐹)=0. Put πœ‡=πœ‡πœ‘,𝑒|πœ•π΅π‘. As in the case of composition operators, it is well known that the boundedness of π‘’πΆπœ‘βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁) impliesπœ‡ξ‚€ξ‚π‘†(𝜁,𝑑)β‰€πΆπ‘‘π‘žπ‘/π‘ξ‚€πœβˆˆπœ•π΅π‘ξ‚,𝑑>0,(2.16)for some positive constant 𝐢 (see [15]). By Lemma 2.2, we see that πœ‡β‰‘0 (if 𝑝<π‘ž) or πœ‡ is absolutely continuous with respect to π‘‘πœŽ (if 𝑝=π‘ž). Thus we haveξ‚€πœ‘0β‰₯πœ‡βˆ—ξ‚β‰‘ξ€œ(𝐸)πœ‘βˆ—βˆ’1(πœ‘βˆ—(𝐸))|||π‘’βˆ—|||π‘žξ€œπ‘‘πœŽβ‰₯𝐸|||π‘’βˆ—|||π‘žπ‘‘πœŽ.(2.17)That is, π‘’βˆ—=0 a.e. on 𝐸. Hence [17, page 83, Theorem 5.5.9] gives that 𝑒≑0 in 𝐡𝑁. This contradicts 𝑒≒0.

Lemma 2.6. Let 1<π‘β‰€π‘ž<∞ and π‘“βˆˆπ»π‘(𝐡𝑁). Suppose that πœ‘βˆΆπ΅π‘β†’π΅π‘ is a holomorphic map and π‘’βˆˆπ»π‘ž(𝐡𝑁)⧡{0} such that π‘’πΆπœ‘βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁) is bounded. Then π‘’βˆ—(𝑓0π‘₯025𝑒6πœ‘)βˆ—=π‘’βˆ—(π‘“βˆ—0π‘₯025𝑒6πœ‘βˆ—) a.e. [𝜎] on πœ•π΅π‘. Here the notation π‘“βˆ— is used as in Lemma 2.3.

Proof. Since πœ‘βˆ— cannot carry a set of positive measure in πœ•π΅π‘ into a set of measure 0 in πœ•π΅π‘ (by Proposition 2.5) and since the radial limit of πœ‘, 𝑓 and πœ“ exist on a set of full measure in πœ•π΅π‘, we have limπ‘Ÿβ†’1βˆ’π‘’βˆ—(π‘“π‘Ÿ0π‘₯025𝑒6πœ‘βˆ—)=π‘’βˆ—(π‘“βˆ—0π‘₯025𝑒6πœ‘βˆ—) a.e. [𝜎] on πœ•π΅π‘.
On the other hand, since π‘“π‘Ÿ is in the ball algebra 𝐴(𝐡𝑁) and π‘“π‘Ÿβ†’π‘“ as π‘Ÿβ†’1βˆ’ in 𝐻𝑝(𝐡𝑁), the boundedness of π‘’πΆπœ‘ shows thatξ€œ0β‰€πœ•π΅π‘|||π‘’βˆ—(𝜁)(𝑓0π‘₯025𝑒6πœ‘)βˆ—(𝜁)βˆ’π‘’βˆ—ξ‚€π‘“(𝜁)βˆ—0π‘₯025𝑒6πœ‘βˆ—ξ‚|||(𝜁)π‘žπ‘‘πœŽ(𝜁)≀liminfπ‘Ÿβ†’1βˆ’ξ€œπœ•π΅π‘|||π‘’βˆ—(𝜁)(𝑓0π‘₯025𝑒6πœ‘)βˆ—(𝜁)βˆ’π‘’βˆ—ξ‚€π‘“(𝜁)π‘Ÿξ‚0π‘₯025𝑒6πœ‘βˆ—|||(𝜁)π‘žπ‘‘πœŽ(𝜁)=liminfπ‘Ÿβ†’1βˆ’β€–β€–π‘’πΆπœ‘π‘“βˆ’π‘’πΆπœ‘π‘“π‘Ÿβ€–β€–π‘žπ»π‘ž=0.(2.18)This implies that π‘’βˆ—(𝑓0π‘₯025𝑒6πœ‘)βˆ—=π‘’βˆ—(π‘“βˆ—0π‘₯025𝑒6πœ‘βˆ—) a.e. [𝜎] on πœ•π΅π‘.

3. Weighted Composition Operators between Hardy Spaces

Theorem 3.1. Let 1<π‘β‰€π‘ž<∞. If π‘’πΆπœ‘ is a bounded weighted composition operator from 𝐻𝑝(𝐡𝑁) into π»π‘ž(𝐡𝑁), then β€–β€–π‘’πΆπœ‘β€–β€–π‘žπ‘’,π»π‘β†’π»π‘ž||∼limsup𝑀||β†’1βˆ’ξ€œπœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2|||ξ‚¬πœ‘1βˆ’βˆ—ξ‚­|||(𝜁),𝑀2}π‘žπ‘/π‘π‘‘πœŽ(𝜁)∼limsup𝑑→0+supπœβˆˆπœ•π΅π‘πœ‡πœ‘,𝑒𝑆(𝜁,𝑑)π‘‘π‘žπ‘/𝑝.(3.1)

Proof of the lower estimates. For each π‘€βˆˆπ΅π‘, we define the function 𝑓𝑀 on 𝐡𝑁 by𝑓𝑀||𝑀||(𝑧)={1βˆ’21βˆ’βŸ¨π‘§,π‘€βŸ©2}𝑁/𝑝.(3.2)Then the functions {π‘“π‘€βˆΆπ‘€βˆˆπ΅π‘} belong to the ball algebra 𝐴(𝐡𝑁) and form a bounded sequence of 𝐻𝑝(𝐡𝑁). Take a compact operator π’¦βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁) arbitrarily. Since the bounded sequence {𝑓𝑀} converges to 0 uniformly on compact subsets of 𝐡𝑁 as |𝑀|β†’1βˆ’, we have β€–π’¦π‘“π‘€β€–π»π‘žβ†’0 as |𝑀|β†’1βˆ’. Thus we obtainβ€–β€–π‘’πΆπœ‘β€–β€–βˆ’π’¦π»π‘β†’π»π‘ž||β‰₯𝐢limsup𝑀||β†’1βˆ’β€–β€–ξ‚€π‘’πΆπœ‘ξ‚π‘“βˆ’π’¦π‘€β€–β€–π»π‘ž||β‰₯𝐢limsup𝑀||β†’1βˆ’β€–β€–π‘’πΆπœ‘π‘“π‘€β€–β€–π»π‘ž.(3.3)By the definition of 𝑓𝑀, we also see thatβ€–β€–π‘’πΆπœ‘π‘“π‘€β€–β€–π‘žπ»π‘ž=ξ€œπœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2|||ξ‚¬πœ‘1βˆ’βˆ—ξ‚­|||(𝜁),𝑀2}π‘žπ‘/π‘π‘‘πœŽ(𝜁).(3.4)Combining this with (3.3), we getβ€–β€–π‘’πΆπœ‘β€–β€–βˆ’π’¦π‘žπ»π‘β†’π»π‘ž||β‰₯𝐢limsup𝑀||β†’1βˆ’ξ€œπœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2|||ξ‚¬πœ‘1βˆ’βˆ—ξ‚­|||(𝜁),𝑀2}π‘žπ‘/π‘π‘‘πœŽ(𝜁).(3.5)Since this holds for every compact operator 𝒦, it follows thatβ€–β€–π‘’πΆπœ‘β€–β€–π‘žπ‘’,π»π‘β†’π»π‘ž||β‰₯𝐢limsup𝑀||β†’1βˆ’ξ€œπœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2|||ξ‚¬πœ‘1βˆ’βˆ—ξ‚­|||(𝜁),𝑀2}π‘žπ‘/π‘π‘‘πœŽ(𝜁).(3.6)
Furthermore, we put 𝑀=(1βˆ’π‘‘)𝜁 for each 𝑑,0<𝑑<1 and πœβˆˆπœ•π΅π‘ in the definition of 𝑓𝑀. Since we see that |𝑓(1βˆ’π‘‘)𝜁(𝑧)|β‰₯πΆπ‘‘βˆ’π‘žπ‘/𝑝 for all π‘§βˆˆπ‘†(𝜁,𝑑), we have𝐢supπœβˆˆπœ•π΅π‘πœ‡πœ‘,𝑒𝑆(𝜁,𝑑)π‘‘π‘žπ‘/𝑝≀supπœβˆˆπœ•π΅π‘ξ€œπ‘†(𝜁,𝑑)|||𝑓(1βˆ’π‘‘)𝜁|||π‘žπ‘‘πœ‡πœ‘,𝑒≀supπœβˆˆπœ•π΅π‘β€–β€–π‘’πΆπœ‘π‘“(1βˆ’π‘‘)πœβ€–β€–π‘žπ»π‘ž.(3.7)Letting 𝑑→0+, we get𝐢limsup𝑑→0+supπœβˆˆπœ•π΅π‘πœ‡πœ‘,𝑒𝑆(𝜁,𝑑)π‘‘π‘žπ‘/𝑝≀limsup𝑑→0+supπœβˆˆπœ•π΅π‘β€–β€–π‘’πΆπœ‘π‘“(1βˆ’π‘‘)πœβ€–β€–π‘žπ»π‘ž.(3.8)Combining this with (3.6), we obtain𝐢limsup𝑑→0+supπœβˆˆπœ•π΅π‘πœ‡πœ‘,𝑒𝑆(𝜁,𝑑)π‘‘π‘žπ‘/π‘β‰€β€–β€–π‘’πΆπœ‘β€–β€–π‘žπ‘’,π»π‘β†’π»π‘ž,(3.9)completing the proof of the lower estimates.

To prove the upper estimates, we need some technical results about the polynomial approximation of π‘“βˆˆπ»π‘(𝐡𝑁). Recall that a holomorphic function 𝑓 in 𝐡𝑁 has the homogeneous expansion𝑓(𝑧)=βˆžξ“π‘˜=0||𝛾||=π‘˜π‘(𝛾)𝑧𝛾,(3.10)where 𝛾=(𝛾1,…,𝛾𝑁) is a multi-index, |𝛾|=𝛾1+β‹―+𝛾𝑁, and 𝑧𝛾=𝑧1𝛾1⋯𝑧𝑁𝛾𝑁. For the homogeneous expansion of 𝑓 and any integer 𝑛β‰₯1, let𝑅𝑛𝑓(𝑧)=βˆžξ“π‘˜=𝑛||𝛾||=π‘˜π‘(𝛾)𝑧𝛾,(3.11)and 𝐾𝑛=πΌβˆ’π‘…π‘›, where 𝐼𝑓=𝑓 is the identity operator.

Proposition 3.2. Suppose that 𝑋 is a Banach space of holomorphic functions in 𝐡𝑁 with the property that the polynomials are dense in 𝑋. Then β€–πΎπ‘›π‘“βˆ’π‘“β€–π‘‹β†’0 as π‘›β†’βˆž if and only if sup{β€–πΎπ‘›β€–βˆΆπ‘›β‰₯1}<∞.

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 1<𝑝<∞, then β€–πΎπ‘›π‘“βˆ’π‘“β€–π»π‘β†’0 as π‘›β†’βˆž for each π‘“βˆˆπ»π‘(𝐡𝑁).

Proof. For each π‘“βˆˆπ»π‘(𝐡𝑁) and π‘Ÿ,0<π‘Ÿ<1, the slice function (π‘“π‘Ÿ)𝜁(πœβˆˆπœ•π΅π‘) of π‘“π‘Ÿ is in the disc algebra 𝐴(𝔻). Here, π‘“π‘Ÿ denotes the dilated function of 𝑓, that is π‘“π‘Ÿ(𝑧)=𝑓(π‘Ÿπ‘§). Hence [20, Corollary 3 and Proposition 1] implies that there is a positive constant 𝐢𝑝 such that1ξ€œ2πœ‹πœ‹βˆ’πœ‹|||πΎπ‘›ξ‚€π‘“π‘Ÿξ‚πœξ‚€π‘’π‘–πœƒξ‚|||π‘π‘‘πœƒβ‰€πΆπ‘1ξ€œ2πœ‹πœ‹βˆ’πœ‹|||ξ‚€π‘“π‘Ÿξ‚πœξ‚€π‘’π‘–πœƒξ‚|||π‘π‘‘πœƒ,(3.12)for every integer 𝑛β‰₯1. Since 𝐾𝑛(π‘“π‘Ÿ)𝜁(π‘’π‘–πœƒ)=𝐾𝑛𝑓(π‘Ÿπ‘’π‘–πœƒπœ), integration by slices (see [17, page 15, Proposition 1.4.7.]) showsξ€œπœ•π΅π‘|||𝐾𝑛|||𝑓(π‘Ÿπœ)π‘π‘‘πœŽ(𝜁)β‰€πΆπ‘ξ€œπœ•π΅π‘||||||𝑓(π‘Ÿπœ)π‘π‘‘πœŽ(𝜁),(3.13)that is, ‖𝐾𝑛‖≀𝐢𝑝1/𝑝 for every integer 𝑛β‰₯1. By Proposition 3.2, we see β€–πΎπ‘›π‘“βˆ’π‘“β€–π»π‘β†’0 as π‘›β†’βˆž. This completes the proof of the proposition.

Corollary 3.4. If 1<𝑝<∞, then 𝑅𝑛 converges to 0 pointwise in 𝐻𝑝(𝐡𝑁) as π‘›β†’βˆž. Moreover, sup{β€–π‘…π‘›β€–βˆΆπ‘›β‰₯1}<∞.

Proof. Since 𝑅𝑛𝑓=π‘“βˆ’πΎπ‘›π‘“, Proposition 3.3 shows that ‖𝑅𝑛𝑓‖𝑝→0 as π‘›β†’βˆž. Furthermore, the principle of uniform boundedness implies that sup𝑛β‰₯1‖𝑅𝑛‖<∞.

Lemma 3.5. Let 1<𝑝<∞. For each π‘“βˆˆπ»π‘(𝐡𝑁) and 𝑛β‰₯1, |||𝑅𝑛|||≀‖‖𝑓‖‖𝑓(𝑧)π»π‘βˆžξ“π‘˜=𝑛Γ(𝑁+π‘˜)||𝑧||π‘˜!Ξ“(𝑁)π‘˜.(3.14)

Proof. Let 𝐾𝑀 be the reproducing kernel for 𝐻2(𝐡𝑁) and let 𝐢[𝑓] be the Cauchy-SzegΓΆ projection. Then, the orthogonality of monomials πœπ›Ό implies that𝑅𝑛𝑅𝑓(𝑧)=πΆπ‘›π‘“ξ‚„ξ€œ(𝑧)=πœ•π΅π‘π‘…π‘›π‘“(𝜁)πΎπ‘§ξ€œ(𝜁)π‘‘πœŽ(𝜁)=πœ•π΅π‘π‘“(𝜁)𝑅𝑛𝐾𝑧(𝜁)π‘‘πœŽ(𝜁).(3.15)HΓΆlder's inequality and the expansion of 𝐾𝑧(𝑀) give|||𝑅𝑛|||β‰€ξ€œπ‘“(𝑧)πœ•π΅π‘|||||||||𝑅𝑓(𝜁)𝑛𝐾𝑧|||ξ€œ(𝜁)π‘‘πœŽ(𝜁)≀{πœ•π΅π‘||||||𝑓(𝜁)π‘π‘‘πœŽ(𝜁)}1/𝑝{ξ€œπœ•π΅π‘|||𝑅𝑛𝐾𝑧|||(𝜁)π‘žπ‘‘πœŽ(𝜁)}1/π‘žβ‰€β€–β€–π‘“β€–β€–π»π‘βˆžξ“π‘˜=𝑛Γ(𝑁+π‘˜)||𝑧||π‘˜!Ξ“(𝑁)π‘˜.(3.16)This 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 1<π‘β‰€π‘ž<∞. If π‘’πΆπœ‘ is bounded from 𝐻𝑝(𝐡𝑁) into π»π‘ž(𝐡𝑁), then β€–β€–π‘’πΆπœ‘β€–β€–π‘’,π»π‘β†’π»π‘žβ‰€liminfπ‘›β†’βˆžβ€–β€–π‘’πΆπœ‘π‘…π‘›β€–β€–π»π‘β†’π»π‘ž.(3.17)

Let us prove the upper estimates for the essential norm of π‘’πΆπœ‘.

Proof of the upper estimates. For the sake of convenience, we set𝑀1||=limsup𝑀||β†’1βˆ’βˆ«πœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2|||ξ‚¬πœ‘1βˆ’βˆ—ξ‚­|||(𝜁),𝑀2}ξ€·π‘žπ‘/π‘ξ€Έπ‘€π‘‘πœŽ(𝜁),(3.18)2=limsup𝑑→0+supπœβˆˆπœ•π΅π‘πœ‡πœ‘,𝑒𝑆(𝜁,𝑑)π‘‘π‘žπ‘/𝑝,(3.19)𝐷(𝜁,𝑑)={π‘§βˆˆπ΅π‘βˆΆ||𝑧||𝑧>1βˆ’π‘‘,||𝑧||βˆˆπ‘„(𝜁,𝑑)}.(3.20)πœ€>0𝑅1,0<𝑅1<1
By the notation (3.18), for given ξ€œπœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2|||ξ‚¬πœ‘1βˆ’βˆ—ξ‚­|||(𝜁),𝑀2}π‘žπ‘/π‘π‘‘πœŽ(𝜁)<𝑀1+πœ€,(3.21), we can choose an π‘€βˆˆπ΅π‘ such that|𝑀|β‰₯𝑅1for πœβˆˆπœ•π΅π‘ with 𝑑,0<𝑑≀1βˆ’π‘…1≑𝑑1. For each 𝑀1=(1βˆ’π‘‘)𝜁 and 𝑓𝑀1(𝑧)={(1βˆ’|𝑀1|2)/(1βˆ’βŸ¨π‘§,𝑀1⟩)2}𝑁/𝑝, we put |𝑓𝑀1(𝑧)|𝑝>4βˆ’π‘π‘‘βˆ’π‘. Since the function π‘§βˆˆπ‘†(𝜁,𝑑) satisfies πœ‡πœ‘,𝑒𝑆(𝜁,𝑑)π‘‘π‘žπ‘/π‘ξ€œ<𝐢𝑆(𝜁,𝑑)|||𝑓𝑀1|||(𝑧)π‘žπ‘‘πœ‡πœ‘,𝑒(𝑧)<𝐢(𝑀1+πœ€)(3.22) for all πœβˆˆπœ•π΅π‘, the inequality (3.21) implies that𝑑,0<𝑑≀𝑑1for all 𝑑2,0<𝑑2<1 and all supπœβˆˆπœ•π΅π‘πœ‡πœ‘,𝑒𝑆(𝜁,𝑑)π‘‘π‘žπ‘/𝑝<𝑀2+πœ€(3.23).
By the notation (3.19), we can also choose a 𝑑,0<𝑑≀𝑑2, so thatπœ‡1for all πœ‡2. Let πœ‡πœ‘,𝑒 and 𝐡𝑁⧡(1βˆ’π‘‘1)𝐡𝑁 be the restrictions of 𝐡𝑁⧡(1βˆ’π‘‘2)𝐡𝑁 to πœ‡π‘—(𝑗=1,2) and πœ‡π‘—ξ‚€ξ‚ξ‚€π‘€π‘†(𝜁,𝑑)≀𝐢𝑗𝑑+πœ€π‘žπ‘/𝑝(3.24), respectively. We claim that πœβˆˆπœ•π΅π‘ also satisfies the Carleson measure condition𝑑>0for all 𝑑,0<𝑑≀𝑑𝑗 and 𝑑>𝑑𝑗. By (3.22) or (3.23), these conditions are true for all {𝑄(π‘€π‘˜,𝑑𝑗/3)}. Hence, we assume that π‘€π‘˜βˆˆπ‘„(𝜁,𝑑). For a finite cover 𝑄(𝜁,𝑑)={π‘§βˆˆπœ•π΅π‘βˆΆ|1βˆ’βŸ¨π‘§,𝜁⟩|≀𝑑}, where Ξ“ of the set {𝑄(π‘€π‘˜,𝑑𝑗/3)}, the covering property implies that there exists a disjoint subcollection ξšπ‘„(𝜁,𝑑)βŠ‚Ξ“π‘„ξ‚€π‘€π‘˜,𝑑𝑗.(3.25) of card(Ξ“)≀𝐢(𝑑/𝑑𝑗)𝑁 so thatπœ‡π‘—ξ‚€ξ‚π‘†(𝜁,𝑑)β‰€πœ‡π‘—ξ‚€ξ‚β‰€ξ“π·(𝜁,𝑑)Ξ“πœ‡π‘—ξ‚€π·ξ‚€π‘€π‘˜,π‘‘π‘—β‰€ξ“ξ‚ξ‚Ξ“πœ‡π‘—ξ‚€π‘†ξ‚€π‘€π‘˜,2𝑑𝑗𝑑≀𝐢𝑑𝑗𝑁𝑀𝑗𝑑+πœ€π‘—π‘žπ‘/𝑝𝑀=𝐢𝑗𝑑+πœ€π‘π‘‘π‘—ξ€·π‘ž/π‘βˆ’1𝑁𝑀≀𝐢𝑗𝑑+πœ€π‘žπ‘/𝑝,(3.26)Furthermore, we obtain 𝐢. By the notation (3.20), we have𝑝,π‘žwhere the constant 𝑁 depends only on π‘“βˆˆπ»π‘(𝐡𝑁), and the dimension ‖𝑓‖𝐻𝑝≀1.
Now, we take a function β€–β€–π‘’πΆπœ‘π‘…π‘›π‘“β€–β€–π‘žπ»π‘ž=ξ€œπœ•π΅π‘|||π‘’βˆ—ξ‚€π‘…π‘›π‘“βˆ—0π‘₯025𝑒6πœ‘βˆ—ξ‚|||π‘ž=ξ€œπ‘‘πœŽπ΅π‘|||π‘…π‘›π‘“βˆ—|||π‘žπ‘‘πœ‡πœ‘,𝑒=ξ€œπ΅π‘|||π‘…π‘›π‘“βˆ—|||π‘žπ‘‘πœ‡π‘—+ξ€œ(1βˆ’π‘‘π‘—)𝐡𝑁|||𝑅𝑛𝑓|||π‘žπ‘‘πœ‡πœ‘,𝑒(3.27) with 𝑛β‰₯1. By Lemma 2.6, we haveξ€œπ΅π‘|||π‘…π‘›π‘“βˆ—|||π‘žπ‘‘πœ‡π‘—ξ‚€π‘€β‰€πΆπ‘—ξ‚β€–β€–π‘…+πœ€π‘›π‘“β€–β€–π‘žπ»π‘β‰€πΆsup𝑛β‰₯1β€–β€–π‘…π‘›β€–β€–π‘žξ‚€π‘€π‘—ξ‚+πœ€.(3.28)for all integers ξ€œ(1βˆ’π‘‘π‘—)𝐡𝑁|||𝑅𝑛𝑓|||π‘žπ‘‘πœ‡πœ‘,π‘’β‰€β€–β€–π‘“β€–β€–π‘žπ»π‘{βˆžξ“π‘˜=𝑛Γ(𝑁+π‘˜)|||π‘˜!Ξ“(𝑁)1βˆ’π‘‘π‘—|||π‘˜}π‘žβ€–β€–π‘’β€–β€–π‘žπ»π‘ž.(3.29). Condition (3.24) and Lemma 2.3 implies thatπ‘’πΆπœ‘On the other hand, by Lemma 3.5, we haveπ‘’βˆˆπ»π‘ž(𝐡𝑁)The boundedness of βˆ‘(Ξ“(𝑁+π‘˜)/π‘˜!Ξ“(𝑁))|1βˆ’π‘‘π‘—|π‘˜ implies that βˆžξ“π‘˜=𝑛Γ(𝑁+π‘˜)|||π‘˜!Ξ“(𝑁)1βˆ’π‘‘π‘—|||π‘˜βŸΆ0asπ‘›βŸΆβˆž.(3.30) and the convergence of the series ξ€œ(1βˆ’π‘‘π‘—)𝐡𝑁|||𝑅𝑛𝑓|||π‘žπ‘‘πœ‡π‘’,πœ‘βŸΆ0asπ‘›βŸΆβˆž.(3.31) implies thatβ€–β€–π‘’πΆπœ‘β€–β€–π‘žπ‘’,π»π‘β†’π»π‘žβ‰€liminfπ‘›β†’βˆžβ€–β€–π‘’πΆπœ‘π‘…π‘›β€–β€–π‘žπ»π‘β†’π»π‘žβ‰€πΆsup𝑛β‰₯1β€–β€–π‘…π‘›β€–β€–π‘žξ‚€π‘€π‘—ξ‚+πœ€.(3.32)So we obtainsup𝑛β‰₯1‖𝑅𝑛‖<∞Combining (3.27), (3.28), and (3.31) with Lemma 3.6, we haveπœ€>0Since Corollary 3.4 implies that β€–β€–π‘’πΆπœ‘β€–β€–π‘žπ‘’,π»π‘β†’π»π‘žβ‰€βŽ§βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩||𝐢limsup𝑀||β†’1βˆ’ξ€œπœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2|||ξ‚¬πœ‘1βˆ’βˆ—ξ‚­|||(𝜁),𝑀2}π‘žπ‘/π‘π‘‘πœŽ(𝜁),𝐢limsup𝑑→0+supπœβˆˆπœ•π΅π‘πœ‡πœ‘,𝑒𝑆(𝜁,𝑑)π‘‘π‘žπ‘/𝑝,(3.33), and 1<π‘β‰€π‘ž<∞ was arbitrary, we conclude thatπ‘’πΆπœ‘βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁)which were to be proved.

Corollary 3.7 (see [15]). Suppose that π‘’πΆπœ‘βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁). For the bounded weighted composition operator 𝑒, the following conditions are equivalent:
(a)πœ‘ is compact;(b)||lim𝑀||β†’1βˆ’ξ€œπœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2|||ξ‚¬πœ‘1βˆ’βˆ—ξ‚­|||(𝜁),𝑀2}π‘žπ‘/π‘π‘‘πœŽ(𝜁)=0;(3.34) and 𝑒 satisfyπœ‘(c)lim𝑑→0+supπœβˆˆπœ•π΅π‘πœ‡πœ‘,𝑒𝑆(𝜁,𝑑)π‘‘π‘žπ‘/𝑝=0.(3.35) and 𝑀𝑒 satisfy1<π‘β‰€π‘ž<∞

4. Multiplication Operators between Hardy Spaces

In this section, we consider the compact multiplication operator π‘€π‘’βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁) between Hardy spaces. As a consequence of Theorem 3.1, we obtain the following results.

Corollary 4.1. Suppose that β€–β€–π‘€π‘’β€–β€–π‘žπ‘’,π»π‘β†’π»π‘ž||∼limsup𝑀||β†’1βˆ’ξ€œπœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2||||||1βˆ’βŸ¨πœ,π‘€βŸ©2}π‘žπ‘/π‘π‘‘πœŽ(𝜁).(4.1). For the bounded multiplication operator π‘€π‘’βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁), the following inequality holds: ||lim𝑀||β†’1βˆ’ξ€œπœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘ž{||𝑀||1βˆ’2||||||1βˆ’βŸ¨πœ,π‘€βŸ©2}π‘žπ‘/π‘π‘‘πœŽ(𝜁)=0.(4.2)Furthermore, 𝑀𝑒 is compact if and only if 𝐻𝑝(𝐡𝑁)

By using Corollary 4.1, we can completely characterize the compactness of a multiplication operator π»π‘ž(𝐡𝑁) from 1<π‘β‰€π‘ž<∞ into π‘€π‘’βˆΆπ»π‘(𝐡𝑁)β†’π»π‘ž(𝐡𝑁).

Theorem 4.2. Suppose that 𝑒=0. Then 𝐡𝑁 is compact if and only if 𝑒≑0 in 𝑀𝑒.

Proof. If 𝑀𝑒, then 𝑒≑0 is compact. Thus, we only prove that the compactness of 𝑀𝑒 implies π‘’βˆˆπ»π‘ž(𝐡𝑁). The boundedness of 𝑒 implies that ξ€œπ‘’(𝑀)=πœ•π΅π‘π‘’βˆ—ξ‚€(𝜁)𝑃(𝑀,𝜁)π‘‘πœŽ(𝜁)π‘€βˆˆπ΅π‘ξ‚,(4.3). Hence, the Poisson representation for 𝑃(𝑀,𝜁) gives that||||β‰€ξ€œπ‘’(𝑀)πœ•π΅π‘|||π‘’βˆ—|||ξ€œ(𝜁)𝑃(𝑀,𝜁)π‘‘πœŽ(𝜁)≀{πœ•π΅π‘|||π‘’βˆ—|||(𝜁)π‘žπ‘ƒ(𝑀,𝜁)π‘ž/π‘π‘‘πœŽ(𝜁)}1/π‘ž{ξ€œπœ•π΅π‘ξ€·π‘ƒ(𝑀,𝜁)1βˆ’1/π‘ξ€Έπ‘žβ€²π‘‘πœŽ(𝜁)}1/π‘žβ€²,(4.4)where 1/π‘ž+1/π‘žβ€²=1 is the Poisson kernel. HΓΆlder's inequality shows that1<π‘β‰€π‘ž<∞where ξ‚€1𝑠≑1βˆ’π‘ξ‚π‘žβ€²=π‘ž(π‘βˆ’1)≀𝑝(π‘žβˆ’1)π‘π‘žβˆ’π‘π‘(π‘žβˆ’1)=1,(4.5). By the assumption ξ€œπœ•π΅π‘ξ€·π‘ƒ(𝑀,𝜁)1βˆ’1/π‘ξ€Έπ‘žβ€²ξ€œπ‘‘πœŽ(𝜁)≀{πœ•π΅π‘{𝑃(𝑀,𝜁)𝑠}1/π‘ π‘‘πœŽ(𝜁)}𝑠=1.(4.6), we see thatlim|𝑀|β†’1βˆ’|𝑒(𝑀)|=0and so we haveπ‘’βˆˆπ»π‘ž(𝐡𝑁)Inequality (4.4) and Corollary 4.1 give that 𝑒. Since 𝐾, this implies that 0 has a 𝜎-limit πœ•π΅π‘ on a set of positive 𝑒≑0-measure in 𝐻∞. Hence [17, page 83, Theorem 5.5.9] shows that 𝛼. This completes the proof.


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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