The essential norm of any operator from a general Banach space of holomorphic functions on the unit ball in ℂ𝑛 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 [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 𝐴1𝛼(𝔻).

We obtain a formula for the essential norm of any operator into a weighted-type space on the unit ball in ℂ𝑛 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 ℂ𝑛 and 𝐻(𝔹) the space of all holomorphic functions on 𝔹. The pseudohyperbolic distance between 𝑧,π‘’βˆˆπ”Ή is denoted by 𝜌(𝑧,𝑒) (see [9] for more details). If 𝑋 is a Banach space, by 𝐡𝑋 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) π»βˆžπ‘£(𝔹)=π»βˆžπ‘£ is defined by π»βˆžπ‘£=ξ‚»π‘“βˆˆπ»(𝔹)βˆΆβ€–π‘“β€–π»βˆžπ‘£βˆΆ=supπ‘§βˆˆπ”Ή||||ξ‚Ό.𝑣(𝑧)𝑓(𝑧)<∞(1.1) The little weighted-type space 𝐻0𝑣(𝔹)=𝐻0𝑣 consists of all π‘“βˆˆπ»βˆžπ‘£ such that lim|𝑧|β†’1||||𝑣(𝑧)𝑓(𝑧)=0.(1.2) With the norm β€–β‹…β€–π»βˆžπ‘£, both are Banach spaces. The norm topology of π»βˆžπ‘£ is finer than the topology 𝜏 of uniform convergence on compact subsets of 𝔹. For 𝑣𝛼(𝑧)=(1βˆ’|𝑧|2)𝛼, 𝛼>0, the standard weighted-type spaces, which we denote by π»βˆžπ›Ό and 𝐻0𝛼, are obtained. These spaces appear in the study of growth conditions of analytic functions, see, for example, [10, 11].

The Bloch-type space β„¬πœ‡(𝔹)=β„¬πœ‡ consists of all π‘“βˆˆπ»(𝔹) such that π‘πœ‡(𝑓)=supπ‘§βˆˆπ”Ή||||πœ‡(𝑧)β„œπ‘“(𝑧)<∞,(1.3) where πœ‡ is a weight, β„œπ‘“(𝑧)=𝑛𝑗=1π‘§π‘—πœ•π‘“πœ•π‘§π‘—ξ«(𝑧)=βˆ‡π‘“(𝑧),𝑧(1.4) is the radial derivative of 𝑓, and βˆ‡π‘“ is the complex gradient of 𝑓.

The little Bloch-type space β„¬πœ‡,0(𝔹)=β„¬πœ‡,0 consists of all π‘“βˆˆβ„¬πœ‡ such that lim|𝑧|β†’1||||πœ‡(𝑧)β„œπ‘“(𝑧)=0.(1.5) Both are Banach spaces with the norm β€–π‘“β€–β„¬πœ‡=|𝑓(0)|+π‘πœ‡(𝑓). For the standard weight 𝑣𝑝, 𝑝>0, we get the 𝑝-Bloch space ℬ𝑝 and the little 𝑝-Bloch space ℬ𝑝0.

The standard weighted-Bergman space 𝐴𝑝𝛽(𝔹)=𝐴𝑝𝛽, 𝛽>βˆ’1, 𝑝β‰₯1, is the set of all analytic functions on 𝔹 such that ‖𝑓‖𝑝𝐴𝑝𝛽=ξ€œπ”Ή||||𝑓(𝑧)𝑝𝑐𝛽1βˆ’|𝑧|2𝛽𝑑𝑉(𝑧)<∞,(1.6) where 𝑑𝑉(𝑧) is the normalized volume measure on 𝔹 and 𝑐𝛽=Ξ“(𝑛+𝛽+1)/𝑛!Ξ“(𝛽+1).

A weight 𝑣 is radial if it satisfies 𝑣(𝑧)=𝑣(|𝑧|) for every π‘§βˆˆπ”Ή. 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|β†’1βˆ’π‘£(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 Μƒξ€·ξ€½||||𝑣(𝑧)∢=sup𝑓(𝑧)βˆΆπ‘“βˆˆπ»βˆžπ‘£,β€–π‘“β€–π»βˆžπ‘£β‰€1ξ€Ύξ€Έβˆ’1.(1.7) For a typical weight 𝑣 the associated weight ̃𝑣 is also typical. Furthermore, for each π‘§βˆˆπ”Ή there is an π‘“π‘§βˆˆπ»βˆžπ‘£, β€–π‘“π‘§β€–π»βˆžπ‘£β‰€1, such that 𝑓𝑧̃(𝑧)=1/𝑣(𝑧), and the same holds for the space 𝐻0𝑣. It is known that π»βˆžΜƒπ‘£β‰…π»βˆžπ‘£ and 𝐻0̃𝑣≅𝐻0𝑣, that is, they are isometrically isometric [10].

We say that a weight 𝑣 satisfies condition 𝐿1 if it is radial and infπ‘˜βˆˆβ„•π‘£ξ€·1βˆ’2βˆ’(π‘˜+1)𝑣1βˆ’2βˆ’π‘˜ξ€Έ>0.(𝐿1) For radial weights 𝑣 satisfying condition 𝐿1, we have that 𝑣 and ̃𝑣 are equivalent, that is, there is a 𝐢β‰₯1 such that ̃𝑣≀𝑣≀𝐢𝑣. Recently Lusky and Taskinen [12] have shown, among other results, that 𝐻0𝛼 is isomorphic to 𝑐0.

Since the closed unit ball π΅π»βˆžπ‘£ is a compact subset of (𝐻(𝔹),𝜏), a result of Dixmier-Ng [13] gives that the subspace of (π»βˆžπ‘£)βˆ—πΊβˆžπ‘£ξ€½ξ€·π»βˆΆ=π‘™βˆˆβˆžπ‘£ξ€Έβˆ—βˆΆπ‘™βˆ£π΅π»βˆžπ‘£ξ€Ύis𝜏-continuous(1.8) is a predual of π»βˆžπ‘£, that is, (πΊβˆžπ‘£)βˆ—β‰…π»βˆžπ‘£. Clearly the evaluation functional at π‘§βˆˆπ”Ή, defined by 𝛿𝑧(𝑓)=𝑓(𝑧), belongs to πΊβˆžπ‘£. The norm of 𝛿𝑧 is denoted by ‖𝛿𝑧‖𝑣. Moreover, the set {π›Ώπ‘§βˆΆπ‘§βˆˆπ”Ή} is a total set, that is, its linear span is norm dense in πΊβˆžπ‘£. More precisely, the next isomorphism result is due to Bierstedt and Summers.

Lemma 1.1 (see [11]). The map 𝑓↦[π‘™β†¦βŸ¨π‘™,π‘“βŸ©] is an onto isometric isomorphism between π»βˆžπ‘£ and (πΊβˆžπ‘£)βˆ— and the restriction map 𝑙↦𝑙|𝐻0𝑣 gives rise to an isometric isomorphism between πΊβˆžπ‘£ and (𝐻0𝑣)βˆ—.

Similarly to the corresponding result in the one variable [14], one can prove the following.

Lemma 1.2. Suppose 𝛽>βˆ’1 and 𝛾>0. Then (𝐴1𝛽)βˆ— is isomorphic to 𝐻0π›ΎβŸ¨π‘“,π‘”βŸ©π›½,𝛾=ξ€œπ”Ήπ‘“(𝑧)𝑔(𝑧)𝑐𝛽+π›Ύπœπ›½+𝛾(𝑧)𝑑𝑉(𝑧),π‘“βˆˆπ»βˆžπ›Ύ,π‘”βˆˆπ΄1𝛽.(1.9) Moreover, under the pairing βŸ¨π‘“,π‘”βŸ©π›½,𝛾 with π‘”βˆˆπ»0𝛾 and π‘“βˆˆπ΄1𝛽, we also have that (𝐻0𝛾)βˆ— is isomorphic to 𝐴1𝛽.

For 𝑧,π‘€βˆˆπ”Ή, let 𝐾𝛽𝑧1(𝑀)=(1βˆ’βŸ¨π‘€,π‘§βŸ©)𝑛+𝛽+1.(1.10) Then the kernel function 𝐾𝑧𝛽+𝛾 clearly belongs to 𝐻0𝛾 and to 𝐴1𝛽. It has the reproducing property ξ€œπ‘“(𝑧)=𝔹𝑓(𝑀)𝐾𝛽𝑧(𝑀)π‘π›½πœπ›½(𝑀)𝑑𝑉(𝑀),π‘§βˆˆπ”Ή,(1.11) for every function π‘“βˆˆπ΄1𝛽 (see [9, Theorem 2.2]). As a direct application we get that 𝑔(𝑀)=βŸ¨πΎπ‘€π›½+𝛾,π‘”βŸ©π›½,𝛾 for all π‘”βˆˆπ΄1𝛽 and 𝑓(𝑀)=βŸ¨π‘“,𝐾𝑀𝛽+π›ΎβŸ©π›½,𝛾 for all π‘“βˆˆπ»βˆžπ›Ύ.

Let 𝐸 be a Banach space of analytic functions on 𝔹 containing the constant functions. We denote its norm by ‖⋅‖𝐸. With β€–β‹…β€–πΈβˆ— we denote the norm of its Banach dual space πΈβˆ—. Consider the following conditions on 𝐸.(𝐢1)There are positive constants 𝑠 and 𝐢 such that ||||𝑓(𝑧)≀𝐢‖𝑓‖𝐸1βˆ’|𝑧|2𝑠,foreveryπ‘“βˆˆπΈ,andforeachπ‘§βˆˆπ”Ή.(1.12)(𝐢2)The analytic polynomials are dense on 𝐸.(𝐢3)β€–π›Ώπ‘§β€–πΈβˆ—β†’βˆž, when |𝑧|β†’1.(𝐢4)The linear span of the set {π›Ώπ‘§βˆΆπ‘§βˆˆπ”Ή} is dense in πΈβˆ—.(𝐢5)There is a 𝐢>0 such that β€–π‘“β€–πΈβ‰€πΆβ€–π‘“β€–βˆž for all π‘“βˆˆπΈ, where β€–π‘“β€–βˆž=supπ‘§βˆˆπ”Ή|𝑓(𝑧)|, the supremum taken in the extended real line.

It follows from (𝐢1) that the closed unit ball 𝐡𝐸 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 (𝐢1) and (𝐢5). Then the contraction operators πΎπ‘ŸβˆΆπΈβ†’πΈ given by πΎπ‘Ÿ(𝑓)(𝑧)∢=𝑓(π‘Ÿπ‘§),0<π‘Ÿ<1,(1.13) are well defined and compact.

Proof. Each given π‘“βˆˆπΈ may be approximated uniformly on compact subsets by the sequence (π‘†π‘˜)π‘˜βˆˆβ„•0 of its Taylor polynomials at 0. Therefore, π‘†π‘˜(π‘Ÿπ‘§)→𝑓(π‘Ÿπ‘§) uniformly on 𝔹 as π‘˜β†’βˆž. Further, (π‘†π‘˜(π‘Ÿπ‘§))π‘˜βˆˆβ„•0βŠ‚πΈ is a Cauchy sequence by (𝐢5), so it converges to some element in 𝐸. Hence πΎπ‘Ÿ(𝑓)∈𝐸.
Moreover πΎπ‘Ÿ is compact. Indeed, any sequence (π‘“π‘š)π‘šβˆˆβ„•βŠ‚π΅πΈ has a 𝜏-convergent subsequence, say itself, to an element π‘”βˆˆπ»(𝔹). Therefore (π‘“π‘š(π‘Ÿπ‘§))π‘šβˆˆβ„• converges uniformly on 𝔹 to 𝑔(π‘Ÿπ‘§), and again (𝐢5) yields that (πΎπ‘Ÿ(π‘“π‘š))π‘šβˆˆβ„• is a Cauchy sequence in 𝐸 that converges to πΎπ‘Ÿ(𝑔).

We consider another condition on 𝐸.(𝐢6)For every π‘“βˆˆπΈ, we have β€–πΎπ‘Ÿ(𝑓)‖𝐸≀‖𝑓‖𝐸 for 0<π‘Ÿ<1.

Remarks 1. (a) Note that (𝐢6) implies that β€–πΎπ‘Ÿβ€–πΈβ†’πΈ=sup‖𝑓‖𝐸≀1β€–πΎπ‘Ÿ(𝑓)‖𝐸≀1.
(b) By (𝐢1) every functional at π‘§βˆˆπ”Ήπ‘›, π›Ώπ‘§βˆΆπΈβ†’β„‚, is bounded, and therefore||𝛿𝑧||≀‖‖𝛿(𝑓)π‘§β€–β€–πΈβˆ—β€–π‘“β€–πΈ,foreveryπ‘“βˆˆπΈ.(1.14)
 (c) Spaces like 𝐻0𝑣, the little Bloch space, Hardy spaces 𝐻𝑝 and weighted-Bergman spaces 𝐴𝑝𝛽 fulfill conditions (𝐢1)–(𝐢6). For 𝐻𝑝, 𝑠=𝑛/𝑝, and for 𝐴𝑝𝛽, we have that 𝑠=(𝑛+𝛽+1)/𝑝 (see [9]).

Let 𝑋 and π‘Œ 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, ‖𝑇‖𝑒=inf‖𝑇+πΎβ€–π‘‹β†’π‘Œξ€ΎβˆΆπΎiscompact.(1.15)

We write 𝐴βͺ―𝐡 if there is a positive constant 𝐢, not depending on properties of 𝐴 and 𝐡, such that 𝐴≀𝐢𝐡. We also write 𝐴≃𝐡 whenever 𝐴βͺ―𝐡 and 𝐡βͺ―𝐴.

2. 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 (𝐢1)–(𝐢6). Then there exists a sequence (πΏπ‘š)π‘šβˆˆβ„• of compact operators on 𝐸 such that (i)for any 0<𝑑<1, limπ‘šβ†’βˆžsup‖𝑓‖𝐸≀1sup|𝑧|≀𝑑||ξ€·πΌβˆ’πΏπ‘šξ€Έ||(𝑓)(𝑧)=0;(2.1)(ii)limsupπ‘šβ†’βˆžβ€–β€–πΌβˆ’πΏπ‘šβ€–β€–πΈβ†’πΈβ‰€1.(2.2)

Proof. We prove that for every 0<𝑑<1 and πœ€>0 there is a compact operator πΏβˆΆπΈβ†’πΈ such that β€–πΌβˆ’πΏβ€–πΈβ†’πΈ<1+2πœ€,(2.3) and sup‖𝑓‖𝐸≀1sup|𝑧|≀𝑑||||(πΌβˆ’πΏ)𝑓(𝑧)<πœ€.(2.4) When this is done, a standard diagonal argument by taking a sequence (𝑑𝑛)↑1 and a sequence of positive numbers (πœ€π‘›)↓0 will give the result.
The operator 𝐿 will be constructed as a suitable finite convex combination of the operators πΎπ‘Ÿ and therefore by Proposition 1.3, it will be compact.
The operators πΌβˆ’πΎπ‘ŸβˆΆ(𝐻(𝔹),𝜏)β†’(𝐻(𝔹),𝜏) are continuous, and for each π‘“βˆˆπ»(𝔹), we have that (πΌβˆ’πΎπ‘Ÿ)𝑓→0 in (𝐻(𝔹),𝜏) when π‘Ÿβ†’1. By the Banach-Steinhaus theorem for FrΓ©chet spaces (πΌβˆ’πΎπ‘Ÿ)β†’0 uniformly on relatively compact subsets of (𝐻(𝔹),𝜏) as π‘Ÿβ†’1; thus in particular on 𝐡𝐸. Hence, if we pick 𝑠1 such that 𝑑<𝑠1<1, we havelimπ‘Ÿβ†’1sup‖𝑓‖𝐸≀1sup|𝑧|≀𝑠1||ξ€·πΌβˆ’πΎπ‘Ÿξ€Έπ‘“||(𝑧)=0.(2.5)
Since 𝐡𝐸 is an equicontinuous set, the operator π›ΏβˆΆπ”Ήβ†’πΈβˆ—, 𝑧↦𝛿𝑧 is continuous, so we can find 𝑧0βˆˆπ”Ή such that sup|𝑧|≀𝑠1β€–π›Ώπ‘§β€–πΈβˆ—=‖𝛿𝑧0β€–πΈβˆ—. Moreover we have the inequalities1β€–1‖𝐸=||𝛿𝑧||(1)β€–1β€–πΈβ‰€β€–β€–π›Ώπ‘§β€–β€–πΈβˆ—.(2.6)
Therefore, we find an π‘Ÿ1∈(0,1) such thatsup‖𝑓‖𝐸≀1sup|𝑧|≀𝑠1||ξ€·πΌβˆ’πΎπ‘Ÿ1ξ€Έ||𝑓(𝑧)β€–β€–π›Ώπ‘§β€–β€–πΈβˆ—ξƒ―πœ€<minπœ€,‖‖𝛿𝑧0β€–β€–πΈβˆ—ξƒ°.(2.7)
Since the bounded set {𝛿𝑧/β€–π›Ώπ‘§β€–πΈβˆ—βˆΆπ‘§βˆˆπ”Ή} is equicontinuous in πΈβˆ—, 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,lim|𝑧|β†’1ξƒ‘π›Ώπ‘§β€–β€–π›Ώπ‘§β€–β€–πΈβˆ—ξƒ’,𝑃=lim|𝑧|β†’1𝑃(𝑧)β€–β€–π›Ώπ‘§β€–β€–πΈβˆ—=0,(2.8) since condition (𝐢3) holds and 𝑃 is bounded on 𝔹.
This means that lim|𝑧|β†’1𝛿𝑧/β€–π›Ώπ‘§β€–πΈβˆ—=0 on relatively compact sets in 𝐸, in particular on πΎπ‘Ÿ1(𝐡𝐸). Hence there is an 𝑠2>𝑠1 such that for each π‘“βˆˆπ΅πΈ, we havesup|𝑧|β‰₯𝑠2||π›Ώπ‘§ξ€·πΎπ‘Ÿ1𝑓||β€–β€–π›Ώπ‘§β€–β€–πΈβˆ—β‰€πœ€.(2.9) Therefore, sup‖𝑓‖𝐸≀1sup|𝑧|β‰₯𝑠2||ξ€·πΌβˆ’πΎπ‘Ÿ1𝑓||(𝑧)β€–β€–π›Ώπ‘§β€–β€–πΈβˆ—<1+πœ€.(2.10)
We can continue in this way and find two strictly increasing sequences (π‘ π‘˜) and (π‘Ÿπ‘˜) satisfyingsup‖𝑓‖𝐸≀1sup|𝑧|β‰€π‘ π‘˜||ξ€·πΌβˆ’πΎπ‘Ÿπ‘˜ξ€Έ||𝑓(𝑧)β€–π›Ώπ‘§β€–πΈβˆ—ξƒ―πœ€<minπœ€,‖𝛿𝑧0β€–πΈβˆ—ξƒ°,(2.11)sup‖𝑓‖𝐸≀1sup|𝑧|β‰₯π‘ π‘˜+1||ξ€·πΌβˆ’πΎπ‘Ÿπ‘˜ξ€Έ||𝑓(𝑧)β€–π›Ώπ‘§β€–πΈβˆ—<1+πœ€.(2.12) Let π‘š>1/πœ€ and put 1𝐿=π‘šπ‘šξ“π‘˜=1πΎπ‘Ÿπ‘˜.(2.13) Then by (2.11) and the fact that ‖𝛿𝑧0β€–πΈβˆ—β‰₯β€–π›Ώπ‘§β€–πΈβˆ— for |𝑧|≀𝑠1, we have that (2.4) holds since sup‖𝑓‖𝐸≀1sup|𝑧|≀𝑑||||≀1(πΌβˆ’πΏ)𝑓(𝑧)π‘šπ‘šξ“π‘˜=1sup‖𝑓‖𝐸≀1sup|𝑧|≀𝑠1||ξ€·πΌβˆ’πΎπ‘Ÿπ‘˜ξ€Έ||𝑓(𝑧)<πœ€.(2.14)
Now we show that (2.3) holds. Similarly to above, we have thatsup‖𝑓‖𝐸≀1sup|𝑧|<𝑑||||(πΌβˆ’πΏ)𝑓(𝑧)β€–π›Ώπ‘§β€–πΈβˆ—<πœ€.(2.15)
Moreover, by (2.12), if 𝑠𝑙<|𝑧|≀𝑠𝑙+1 and ‖𝑓‖𝐸≀1, then||ξ€·πΌβˆ’πΎπ‘Ÿπ‘˜ξ€Έπ‘“||(𝑧)β€–β€–π›Ώπ‘§β€–β€–πΈβˆ—<1+πœ€,(2.16) except possibly for π‘˜=𝑙, in which case ||ξ€·πΌβˆ’πΎπ‘Ÿπ‘™ξ€Έπ‘“||(𝑧)β€–β€–π›Ώπ‘§β€–β€–πΈβˆ—β‰€β€–β€–(πΌβˆ’πΎπ‘Ÿπ‘™β€–β€–)π‘“πΈβ‰€β€–β€–πΌβˆ’πΎπ‘Ÿπ‘™β€–β€–πΈβ†’πΈβ‰€2.(2.17) Hence, we get for all π‘§βˆˆπ”Ή and ‖𝑓‖𝐸≀1 that ||||(πΌβˆ’πΏ)𝑓(𝑧)β€–π›Ώπ‘§β€–πΈβˆ—β‰€1π‘šξ“π‘˜β‰ π‘™||ξ€·πΌβˆ’πΎπ‘Ÿπ‘˜ξ€Έπ‘“||(𝑧)β€–π›Ώπ‘§β€–πΈβˆ—+1π‘š||ξ€·πΌβˆ’πΎπ‘Ÿπ‘™ξ€Έ||𝑓(𝑧)β€–π›Ώπ‘§β€–πΈβˆ—<1+2πœ€.(2.18) In light of condition (𝐢4), we have β€–πΌβˆ’πΏβ€–πΈβ†’πΈ=supπ‘§βˆˆπ”Ήβ€–β€–β€–(πΌβˆ’πΏ)βˆ—π›Ώπ‘§β€–β€–π›Ώπ‘§β€–β€–πΈβˆ—β€–β€–β€–πΈβˆ—=supπ‘§βˆˆπ”Ήsup‖𝑓‖𝐸≀1||||(πΌβˆ’πΏ)𝑓(𝑧)β€–β€–π›Ώπ‘§β€–β€–πΈβˆ—<1+2πœ€.(2.19) 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 π»βˆžπ‘£ or 𝐻0𝑣. We omit its proof.

Theorem 2.2. Suppose π‘‡βˆΆπΈβ†’π»βˆžπ‘£ is a bounded operator. Then β€–π‘‡β€–πΈβ†’π»βˆžπ‘£=supπ‘§βˆˆπ”Ήβ€–β€–π‘‡βˆ—(𝛿𝑧)β€–β€–πΈβˆ—β€–β€–π›Ώπ‘§β€–β€–π‘£.(2.20)

Since many operators can be written as weighted composition operators we state the following useful result. As usual, π‘’πΆπœ‘(𝑓)=𝑒⋅(π‘“βˆ˜πœ‘) where 𝑒 is an analytic function on 𝔹 and πœ‘ is an analytic self-map of 𝔹.

Corollary 2.3. If π‘’πΆπœ‘βˆΆπΈβ†’π»βˆžπ‘£ is a bounded weighted composition operator, then β€–β€–π‘’πΆπœ‘β€–β€–πΈβ†’π»βˆžπ‘£=supπ‘§βˆˆπ”Ή||||̃‖‖𝛿𝑒(𝑧)𝑣(𝑧)πœ‘(𝑧)β€–β€–πΈβˆ—.(2.21)

Example 2.4. Let 𝐸=𝐴𝑝𝛽, 𝛽>βˆ’1, 𝑝>1 with 1/𝑝+1/π‘ž=1 and ̃𝑣𝛼(𝑧)=𝑣𝛼(𝑧)=(1βˆ’|𝑧|2)𝛼, 𝛼>0. Then, since for all π‘“βˆˆπ΄π‘π›½ and π‘§βˆˆπ”Ή, ||||≀𝑓(𝑧)‖𝑓‖𝐴𝑝𝛽1βˆ’|𝑧|2ξ€Έ(𝑛+1+𝛽)/𝑝,(2.22) and since equality is attained at 𝑓𝑧(𝑀)=1βˆ’|𝑧|2ξ€Έ(𝑛+1+𝛽)/𝑝(1βˆ’βŸ¨π‘€,π‘§βŸ©)2(𝑛+1+𝛽)/𝑝,(2.23) we get β€–β€–π›Ώπœ‘(𝑧)β€–β€–π΄π‘žπ›½=1ξ‚€||||1βˆ’πœ‘(𝑧)2(𝑛+1+𝛽)/𝑝.(2.24) So we get β€–β€–π‘’πΆπœ‘β€–β€–π΄π‘π›½β†’π»βˆžπ›Ό=supπ‘§βˆˆπ”Ήξ€·1βˆ’|𝑧|2𝛼||||𝑒(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)2(𝑛+1+𝛽)/𝑝.(2.25)

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

Theorem 2.5. Assume that 𝐸 satisfies conditions (𝐢1)–(𝐢6). If π‘‡βˆΆπΈβ†’π»0𝑣 is a bounded operator, then ‖𝑇‖𝑒=limsup|𝑧|β†’1βˆ’β€–β€–π‘‡βˆ—(𝛿𝑧)β€–β€–πΈβˆ—β€–β€–π›Ώπ‘§β€–β€–π‘£.(2.26)

Proof. If β„“βˆΆ=limsup|𝑧|β†’1βˆ’β€–β€–π‘‡βˆ—(𝛿𝑧)β€–β€–πΈβˆ—β€–β€–π›Ώπ‘§β€–β€–π‘£,(2.27) then the same reasoning as in the first part of [3, Theorem 3.1] shows that ‖𝑇‖𝑒β‰₯β„“.
To prove the reverse inequality, let (πΏπ‘š)π‘šβˆˆβ„• be the sequence provided by Proposition 2.1. Thenβ€–π‘‡β€–π‘’β‰€β€–β€–π‘‡βˆ’π‘‡βˆ˜πΏπ‘šβ€–β€–πΈβ†’π»0𝑣,(2.28) for all π‘šβˆˆβ„•.
Next, we estimatelimsupπ‘šβ†’βˆžβ€–β€–π‘‡βˆ’π‘‡βˆ˜πΏπ‘šβ€–β€–πΈβ†’π»0𝑣.(2.29)
Fix 0<𝑑<1 and 𝑓 in the unit ball of 𝐸. Then||ξ€Ίξ€·π‘‡βˆ˜πΌβˆ’πΏπ‘š||=||𝛿(𝑓)(𝑧)𝑧,ξ€Ίξ€·π‘‡βˆ˜πΌβˆ’πΏπ‘šξ¬||=||ξ€Έξ€»(𝑓)ξ«ξ€·πΌβˆ’πΏβˆ—π‘šξ€Έβˆ˜π‘‡βˆ—ξ€·π›Ώπ‘§ξ€Έξ¬||,𝑓.(2.30)
The sequence (πΌβˆ’πΏβˆ—π‘š)π‘šβˆˆβ„•βŠ‚β„’(πΈβˆ—) is bounded because of (2.2) in Proposition 2.1. Since for each fixed π‘§βˆˆπ”Ή, and using (2.1) in Proposition 2.1limπ‘šβ†’βˆžβ€–β€–(πΌβˆ’πΏβˆ—π‘š)(𝛿𝑧)β€–β€–πΈβˆ—=limπ‘šβ†’βˆžsup‖𝑓‖𝐸≀1||ξ€·πΌβˆ’πΏπ‘šξ€Έ||(𝑓)(𝑧)=0,(2.31) the sequence (πΌβˆ’πΏβˆ—π‘š)π‘šβˆˆβ„• converges to zero on every point in the total set {π›Ώπ‘§βˆΆπ‘§βˆˆπ”Ή}βŠ‚πΈβˆ— as π‘šβ†’βˆž. So we appeal to the Banach-Steinhaus type theorem [15, III 4.5] to conclude that (πΌβˆ’πΏβˆ—π‘š)π‘šβˆˆβ„• converges to zero uniformly on compact subsets of πΈβˆ—. In particular on the image {π‘‡βˆ—(𝛿𝑧)∢|𝑧|≀𝑑}βŠ‚πΈβˆ— of the compact set {π›Ώπ‘§βˆΆ|𝑧|≀𝑑}βŠ‚πΊβˆžπ‘£, that is, 0=limπ‘šβ†’βˆžsup{|𝑧|≀𝑑}β€–β€–ξ€·πΌβˆ’πΏβˆ—π‘šπ‘‡ξ€Έξ€·βˆ—ξ€·π›Ώπ‘§β€–β€–ξ€Έξ€ΈπΈβˆ—=limπ‘šβ†’βˆžsup{‖𝑓‖𝐸≀1}sup{|𝑧|≀𝑑}||ξ«ξ€·πΌβˆ’πΏβˆ—π‘šξ€Έβˆ˜π‘‡βˆ—ξ€·π›Ώπ‘§ξ€Έξ¬||.,𝑓(2.32) Therefore since the weight ̃𝑣 is bounded, limπ‘šβ†’βˆžsup{‖𝑓‖𝐸≀1}sup{|𝑧|≀𝑑}||ξ€Ίξ€·π‘‡βˆ˜πΌβˆ’πΏπ‘š||̃𝑣(𝑓)(𝑧)(𝑧)=0.(2.33) On the other hand, for |𝑧|>𝑑, and as in the proof of [3, Theorem 3.1] sup{‖𝑓‖𝐸≀1}||ξ€Ίξ€·π‘‡βˆ˜πΌβˆ’πΏπ‘š||Μƒβ€–β€–ξ€Έξ€»(𝑓)(𝑧)𝑣(𝑧)β‰€πΌβˆ’πΏπ‘šβ€–β€–πΈβ†’π»0𝑣⋅sup|𝑧|>π‘‘β€–β€–π‘‡βˆ—(𝛿𝑧)β€–β€–πΈβˆ—β€–β€–π›Ώπ‘§β€–β€–π‘£.(2.34) Therefore, limsupπ‘šβ†’βˆžβ€–β€–π‘‡βˆ’π‘‡βˆ˜πΏπ‘šβ€–β€–πΈβ†’π»0𝑣=limsupπ‘šβ†’βˆžsup{‖𝑓‖𝐸≀1}sup{|𝑧|<1}||ξ€Ίξ€·π‘‡βˆ˜πΌβˆ’πΏπ‘š||̃𝑣(𝑓)(𝑧)(𝑧)=limsupπ‘šβ†’βˆžsup{‖𝑓‖𝐸≀1}sup{|𝑧|>𝑑}||ξ€Ίξ€·π‘‡βˆ˜πΌβˆ’πΏπ‘š||Μƒξ€Έξ€»(𝑓)(𝑧)𝑣(𝑧)≀limsupπ‘šβ†’βˆžβ€–πΌβˆ’πΏπ‘šβ€–πΈβ†’π»0𝑣⋅sup|𝑧|>π‘‘β€–π‘‡βˆ—ξ€·π›Ώπ‘§ξ€Έβ€–πΈβˆ—β€–π›Ώπ‘§β€–π‘£.(2.35) Applying Proposition 2.1 in (2.35) and letting 𝑑→1 we obtain ‖𝑇‖𝑒≀ℓ.

Corollary 2.6. Assume that 𝐸 satisfies conditions (𝐢1)–(𝐢6). If π‘’πΆπœ‘βˆΆπΈβ†’π»0𝑣 is a bounded weighted composition operator, then β€–β€–π‘’πΆπœ‘β€–β€–π‘’=limsup|𝑧|β†’1βˆ’||||̃‖‖𝛿𝑒(𝑧)𝑣(𝑧)πœ‘(𝑧)β€–β€–πΈβˆ—.(2.36)

Proof. We apply Theorem 2.5 and just use that (π‘’πΆπœ‘)βˆ—(𝛿𝑧)=𝑒(𝑧)π›Ώπœ‘(𝑧) and that ‖𝛿𝑧‖𝑣̃=1/𝑣(𝑧).

Example 2.7. If 𝐸=𝐻0𝑀, then from Corollary 2.6 we get the following extension of the one-dimensional result in [5, Theorem 2.2] β€–β€–π‘’πΆπœ‘β€–β€–π‘’=limsup|𝑧|β†’1βˆ’Μƒ||||𝑣(𝑧)𝑒(𝑧).𝑀(πœ‘(𝑧))(2.37)

Corollary 2.8. Let 𝐹 be a Banach space of analytic functions in 𝔹 that is isometrically isomorphic to the bidual of a space 𝐸 satisfying conditions (𝐢1)–(𝐢6). Assume that 𝐸 has the πœ†-metric approximation property or is a dual space. Let π‘‡βˆΆπΉβ†’π»βˆžπ‘£ be a bounded operator such that 𝑇(𝐸)βŠ†π»0𝑣 and whose restrictions to bounded subsets of 𝐹 are 𝜏-𝜏 continuous. Then ‖𝑇‖𝑒≃limsup|𝑧|β†’1βˆ’β€–β€–π‘‡βˆ—(𝛿𝑧)β€–β€–πΉβˆ—β€–β€–π›Ώπ‘§β€–β€–π‘£.(2.38)

Proof. To see that 𝑇=(𝑇|𝐸)βˆ—βˆ—, it suffices to remark that 𝑇 is weak*-weak* continuous, that is, to check that π‘™βˆ˜π‘‡βˆˆπΈβˆ— for all π‘™βˆˆπΊβˆžπ‘£. And for this, it is enough to check that π‘™βˆ˜π‘‡ is weak* continuous on the closed unit ball π΅πΈβˆ—βˆ— of 𝐹=πΈβˆ—βˆ—, a fact that follows from observing that on (𝐡𝐹,π‘€βˆ—) 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 πœ†-metric approximation property or is a dual space, we may use Axler et al. [16] to obtain that ‖𝑇‖𝑒=β€–(𝑇|𝐸)βˆ—β€–π‘’β‰ƒβ€–π‘‡|𝐸‖𝑒. From this and Theorem 2.5 the corollary follows.

Notice that spaces like π»βˆžπ‘£, the Bloch space and 𝐡𝑀𝑂𝐴 satisfy the assumptions on 𝐹 in the above corollary.

Corollary 2.9. Let 𝛽>βˆ’1, 𝛾>0 and assume that π‘‡βˆΆπ΄1𝛽→𝐴1𝛽 is a bounded operator. If π‘‡ξ…ž(𝐻0𝛾)βŠ†π»0𝛾, where π‘‡ξ…žβˆΆπ»βˆžπ›Ύβ†’π»βˆžπ›Ύ is the dual operator with respect to the duality βŸ¨β‹…,β‹…βŸ©π›½,𝛾, then ‖𝑇‖𝑒≃limsup|𝑧|β†’1βˆ’β€–β€–π‘‡ξ‚€πΎπ‘§π›½+𝛾‖‖𝐴1𝛽‖‖𝐾𝑧𝛽+𝛾‖‖𝐴1𝛽.(2.39)

Proof. Note that ‖𝐾𝑧𝛽+𝛾‖𝐴1𝛽≃(1βˆ’|𝑧|2)βˆ’π›Ύ. Let π‘‡ξ…ž|𝐻0𝛾 be the restriction to 𝐻0𝛾 of π‘‡ξ…ž. Then we can apply Theorem 2.5 so that β€–β€–π‘‡ξ…ž|𝐻0𝛾‖‖𝑒=limsup|𝑧|β†’1βˆ’β€–β€–β€–ξ‚€π‘‡ξ…ž|𝐻0π›Ύξ‚βˆ—(𝛿𝑧)β€–β€–β€–(𝐻0𝛾)βˆ—β€–β€–π›Ώπ‘§β€–β€–π‘£π›Ύ.(2.40) For β€–π‘“β€–π»βˆžπ›Ύβ‰€1, π‘“βˆˆπ»0𝛾 and π‘§βˆˆπ”», |||ξ‚€π‘‡ξ…ž|𝐻0π›Ύξ‚βˆ—π›Ώπ‘§|||=||𝑇(𝑓)ξ…ž||=|||𝐾𝑓(𝑧)𝑧𝛽+𝛾,π‘‡ξ…žπ‘“ξ‚­π›½,𝛾|||=|||𝑇𝐾𝑧𝛽+𝛾,𝑓𝛽,𝛾|||.(2.41) Since by [12] 𝐻0𝛾 is isomorphic to 𝑐0, we obtain that ‖𝑇‖𝑒≃‖‖(π‘‡ξ…ž|𝐻0𝛾)βˆ—β€–β€–π‘’β‰ƒβ€–β€–π‘‡ξ…ž|𝐻0𝛾‖‖𝑒,(2.42) as wanted.

Now we apply Theorem 2.5 to bounded operators on the subspaces of 𝑝-Bloch and little 𝑝-Bloch spaces consisting of all 𝑓 such that 𝑓(0)=0. We denote the spaces by ℬ𝑝/β„‚ and ℬ𝑝0/β„‚ correspondingly. Similar subspaces of π»βˆžπœ‡ and 𝐻0πœ‡ are denoted by π»βˆžπœ‡/β„‚ and 𝐻0πœ‡/β„‚ correspondingly.

Observe that for any π‘§βˆˆπ”Ή, ‖𝛿𝑧‖(ℬ𝑝0)βˆ—β‰€1+‖𝛿𝑧‖(ℬ𝑝0/β„‚)βˆ—. Hence ℬ𝑝0/β„‚ being a subspace of ℬ𝑝0 fulfills conditions (𝐢1)–(𝐢6).

Set ̂𝛿𝑧(𝑓)∢=(1βˆ’|𝑧|2)π‘β„œπ‘“(𝑧) for π‘§βˆˆπ”Ή and π‘“βˆˆβ„¬π‘/β„‚.

Corollary 2.10. (a) Let π‘‡βˆΆβ„¬π‘0/ℂ→ℬ𝑝0/β„‚ be a bounded operator. Then ‖𝑇‖𝑒=limsup|𝑧|β†’1βˆ’β€–β€–π‘‡βˆ—(̂𝛿𝑧)β€–β€–(ℬ𝑝0/β„‚)βˆ—.(2.43)
 (b) Let π‘‡βˆΆβ„¬π‘/ℂ→ℬ𝑝/β„‚ be a bounded operator such that ℬ𝑝0/β„‚ is an invariant subspace for 𝑇 and whose restrictions to the bounded subsets are 𝜏-𝜏 continuous. Then ‖𝑇‖𝑒≃limsup|𝑧|β†’1βˆ’β€–β€–π‘‡βˆ—(̂𝛿𝑧)β€–β€–(ℬ𝑝/β„‚)βˆ—.(2.44)

Proof. Let β„œβˆΆβ„¬π‘/β„‚β†’π»βˆžπ‘/β„‚ be defined by β„œ(𝑓)(𝑧)=β„œπ‘“(𝑧). Then both β„œ and β„œβˆΆβ„¬π‘0/ℂ→𝐻0𝑝/β„‚ are onto isometries (see, e.g., [17]). Therefore, if we put 𝐿=β„œβˆ˜π‘‡, we have ‖𝐿‖𝑒=‖𝑇‖𝑒. Since ‖𝛿𝑧‖𝑣𝑝=(1βˆ’|𝑧|2)βˆ’π‘, and β„œ(𝑓)(𝑧)=β„œβˆ—(𝛿𝑧)(𝑓) we also have that πΏβˆ—ξƒ©π›Ώπ‘§β€–β€–π›Ώπ‘§β€–β€–π‘£π‘ξƒͺ=π‘‡βˆ—βˆ˜β„œβˆ—ξƒ©π›Ώπ‘§β€–β€–π›Ώπ‘§β€–β€–π‘£π‘ξƒͺ=π‘‡βˆ—ξ‚€ξ€·1βˆ’|𝑧|2ξ€Έπ‘β„œβˆ—ξ€·π›Ώπ‘§ξ€Έξ‚=π‘‡βˆ—ξ€·Μ‚π›Ώπ‘§ξ€Έ.(2.45) Hence, β€–π‘‡βˆ—(̂𝛿𝑧)β€–(ℬ𝑝0/β„‚)βˆ—=β€–πΏβˆ—(𝛿𝑧/‖𝛿𝑧‖𝑣)β€–(ℬ𝑝0/β„‚)βˆ—.
Now for (a), apply Theorem 2.5 to 𝐿. For (b), observe that 𝐿(ℬ𝑝0/β„‚)βŠ†π»0𝑝 and also that the quotient norm on ℬ𝑝0/β„‚ coincides with 𝑏𝑝, and therefore using the quotient and the subspace duality, (ℬ𝑝0/β„‚)βˆ—βˆ—=ℬ𝑝/β„‚. Moreover, 𝐿 is 𝜏-𝜏 continuous on bounded sets, because β„œ preserves the 𝜏-𝜏 continuity. Then use Corollary 2.8.

For a given π‘”βˆˆπ»(𝔹), 𝑔(0)=0 and a holomorphic self-map πœ‘ of 𝔹, the following integral-type operator on 𝐻(𝔹) is introduced in [17] π‘ƒπ‘”πœ‘ξ€œ(𝑓)(𝑧)=10𝑓(πœ‘(𝑑𝑧))𝑔(𝑑𝑧)𝑑𝑑𝑑,π‘“βˆˆπ»(𝔹),π‘§βˆˆπ”Ή.(2.46)

This operator has also been studied later, for example, in [6, 8, 18, 19]. For a closely related operator see also [20]. In [6] SteviΔ‡ calculated the essential norm of the operator π‘ƒπ‘”πœ‘βˆΆπ΄2π›½β†’β„¬πœ‡. Motivated by this result we calculate here the essential norm of the operator π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›½β†’β„¬πœ‡,0/β„‚, but in terms of the associated weight of πœ‡.

Corollary 2.11. Assume that 𝛽>βˆ’1, 𝑝>1, π‘”βˆˆπ»(𝔹), 𝑔(0)=0, πœ‘ is a holomorphic self-map of 𝔹 and π‘ƒπ‘”πœ‘βˆΆπ΄π‘π›½β†’β„¬πœ‡,0/β„‚ is bounded. Then β€–β€–π‘ƒπ‘”πœ‘β€–β€–π‘’=limsup|𝑧|β†’1||||ξ‚πœ‡(𝑧)𝑔(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)2(𝑛+1+𝛽)/𝑝.(2.47)

Proof. Since β„œπ‘ƒπ‘”πœ‘(𝑓)(𝑧)=𝑓(πœ‘(𝑧))𝑔(𝑧), we have β„œπ‘ƒπ‘”πœ‘=π‘”πΆπœ‘βˆΆπ΄π‘π›½β†’π»0πœ‡/β„‚. From this and the fact that the operator β„œ is an onto isometry between β„¬πœ‡,0/β„‚ and 𝐻0πœ‡/β„‚ it follows that β€–π‘ƒπ‘”πœ‘β€–π‘’=β€–π‘”πΆπœ‘β€–π‘’. By Corollary 2.6, we have β€–β€–π‘”πΆπœ‘β€–β€–π‘’=limsup|𝑧|β†’1βˆ’||||‖‖𝛿𝑔(𝑧)ξ‚πœ‡(𝑧)πœ‘(𝑧)β€–β€–π΄π‘žπ›½,(2.48) where 1/𝑝+1/π‘ž=1. Therefore equality (2.47) follows simply bearing in mind (2.24).


P. Galindo is supported partially by Project MEC-FEDER 2007 064521. M. Lindstrâm 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).