Essential Norm of Operators into Weighted-Type Spaces on the Unit Ball

Abstract

The essential norm of any operator from a general Banach space of holomorphic functions on the unit ball in ℂ^𝑛 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 𝐴^1𝛼(𝔻).

We obtain a formula for the essential norm of any operator into a weighted-type space on the unit ball in ℂ^𝑛 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 ℂ^𝑛 and 𝐻(𝔹) the space of all holomorphic functions on 𝔹. The pseudohyperbolic distance between 𝑧,𝑢∈𝔹 is denoted by 𝜌(𝑧,𝑢) (see [9] for more details). If 𝑋 is a Banach space, by 𝐵_𝑋 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) 𝐻^∞𝑣(𝔹)=𝐻^∞𝑣 is defined by 𝐻^∞𝑣=𝑓∈𝐻(𝔹)∶‖𝑓‖_𝐻^∞𝑣∶=sup_𝑧∈𝔹𝑣(𝑧)||𝑓(𝑧)||<∞.(1.1) The little weighted-type space 𝐻^0𝑣(𝔹)=𝐻^0𝑣 consists of all 𝑓∈𝐻^∞𝑣 such that lim_|𝑧|→1𝑣(𝑧)||𝑓(𝑧)||=0.(1.2) With the norm ‖⋅‖_𝐻^∞𝑣, both are Banach spaces. The norm topology of 𝐻^∞𝑣 is finer than the topology 𝜏 of uniform convergence on compact subsets of 𝔹. For 𝑣_𝛼(𝑧)=(1−|𝑧|²)^𝛼, 𝛼>0, the standard weighted-type spaces, which we denote by 𝐻^∞𝛼 and 𝐻^0𝛼, are obtained. These spaces appear in the study of growth conditions of analytic functions, see, for example, [10, 11].

The Bloch-type space ℬ_𝜇(𝔹)=ℬ_𝜇 consists of all 𝑓∈𝐻(𝔹) such that 𝑏_𝜇(𝑓)=sup_𝑧∈𝔹𝜇(𝑧)||ℜ𝑓(𝑧)||<∞,(1.3) where 𝜇 is a weight, ℜ𝑓(𝑧)=^𝑛_𝑗=1𝑧_𝑗𝜕𝑓𝜕𝑧_𝑗(𝑧)=∇𝑓(𝑧),𝑧(1.4) is the radial derivative of 𝑓, and ∇𝑓 is the complex gradient of 𝑓.

The little Bloch-type space ℬ_𝜇,0(𝔹)=ℬ_𝜇,0 consists of all 𝑓∈ℬ_𝜇 such that lim_|𝑧|→1𝜇(𝑧)||ℜ𝑓(𝑧)||=0.(1.5) Both are Banach spaces with the norm ‖𝑓‖_ℬ𝜇=|𝑓(0)|+𝑏_𝜇(𝑓). For the standard weight 𝑣_𝑝, 𝑝>0, we get the 𝑝-Bloch space ℬ^𝑝 and the little 𝑝-Bloch space ℬ^𝑝0.

The standard weighted-Bergman space 𝐴^𝑝𝛽(𝔹)=𝐴^𝑝𝛽, 𝛽>−1, 𝑝≥1, is the set of all analytic functions on 𝔹 such that ‖𝑓‖^𝑝𝐴_𝑝𝛽=_𝔹||𝑓(𝑧)||^𝑝𝑐_𝛽1−|𝑧|²^𝛽𝑑𝑉(𝑧)<∞,(1.6) where 𝑑𝑉(𝑧) is the normalized volume measure on 𝔹 and 𝑐_𝛽=Γ(𝑛+𝛽+1)/𝑛!Γ(𝛽+1).

A weight 𝑣 is radial if it satisfies 𝑣(𝑧)=𝑣(|𝑧|) for every 𝑧∈𝔹. 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|→1⁻𝑣(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 𝑣̃ is defined as follows 𝑣̃(𝑧)∶=sup||𝑓(𝑧)||∶𝑓∈𝐻^∞𝑣,‖𝑓‖_𝐻^∞𝑣≤1⁻¹.(1.7) For a typical weight 𝑣 the associated weight 𝑣̃ is also typical. Furthermore, for each 𝑧∈𝔹 there is an 𝑓_𝑧∈𝐻^∞𝑣, ‖𝑓_𝑧‖_𝐻^∞𝑣≤1, such that 𝑓_𝑧(𝑧)=1/𝑣(̃𝑧), and the same holds for the space 𝐻^0𝑣. It is known that 𝐻^∞̃𝑣≅𝐻^∞𝑣 and 𝐻^0̃𝑣≅𝐻^0𝑣, that is, they are isometrically isometric [10].

We say that a weight 𝑣 satisfies condition 𝐿1 if it is radial and inf_𝑘∈ℕ𝑣1−2^−(𝑘+1)𝑣1−2^−𝑘>0.(𝐿1) For radial weights 𝑣 satisfying condition 𝐿1, we have that 𝑣 and 𝑣̃ are equivalent, that is, there is a 𝐶≥1 such that 𝑣≤𝑣̃≤𝐶𝑣. Recently Lusky and Taskinen [12] have shown, among other results, that 𝐻^0𝛼 is isomorphic to 𝑐₀.

Since the closed unit ball 𝐵_𝐻^∞𝑣 is a compact subset of (𝐻(𝔹),𝜏), a result of Dixmier-Ng [13] gives that the subspace of (𝐻^∞𝑣)^∗𝐺^∞𝑣∶=𝑙∈𝐻^∞𝑣^∗∶𝑙∣𝐵_𝐻^∞𝑣is𝜏-continuous(1.8) is a predual of 𝐻^∞𝑣, that is, (𝐺^∞𝑣)^∗≅𝐻^∞𝑣. Clearly the evaluation functional at 𝑧∈𝔹, defined by 𝛿_𝑧(𝑓)=𝑓(𝑧), belongs to 𝐺^∞𝑣. The norm of 𝛿_𝑧 is denoted by ‖𝛿_𝑧‖_𝑣. Moreover, the set {𝛿_𝑧∶𝑧∈𝔹} is a total set, that is, its linear span is norm dense in 𝐺^∞𝑣. More precisely, the next isomorphism result is due to Bierstedt and Summers.

Lemma 1.1 (see [11]). The map 𝑓↦[𝑙↦⟨𝑙,𝑓⟩] is an onto isometric isomorphism between 𝐻^∞𝑣 and (𝐺^∞𝑣)^∗ and the restriction map 𝑙↦𝑙|_𝐻^0𝑣 gives rise to an isometric isomorphism between 𝐺^∞𝑣 and (𝐻^0𝑣)^∗.

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

Lemma 1.2. Suppose 𝛽>−1 and 𝛾>0. Then (𝐴^1𝛽)^∗ is isomorphic to 𝐻^0𝛾⟨𝑓,𝑔⟩_𝛽,𝛾=_𝔹𝑓(𝑧)𝑔(𝑧)𝑐_𝛽+𝛾𝜐_𝛽+𝛾(𝑧)𝑑𝑉(𝑧),𝑓∈𝐻^∞𝛾,𝑔∈𝐴^1𝛽.(1.9) Moreover, under the pairing ⟨𝑓,𝑔⟩_𝛽,𝛾 with 𝑔∈𝐻^0𝛾 and 𝑓∈𝐴^1𝛽, we also have that (𝐻^0𝛾)^∗ is isomorphic to 𝐴^1𝛽.

For 𝑧,𝑤∈𝔹, let 𝐾^𝛽𝑧(𝑤)=1(1−⟨𝑤,𝑧⟩)^𝑛+𝛽+1.(1.10) Then the kernel function 𝐾^{𝑧𝛽+𝛾} clearly belongs to 𝐻^0𝛾 and to 𝐴^1𝛽. It has the reproducing property 𝑓(𝑧)=_𝔹𝑓(𝑤)𝐾^𝛽𝑧(𝑤)𝑐_𝛽𝜐_𝛽(𝑤)𝑑𝑉(𝑤),𝑧∈𝔹,(1.11) for every function 𝑓∈𝐴^1𝛽 (see [9, Theorem 2.2]). As a direct application we get that 𝑔(𝑤)=⟨𝐾^{𝑤𝛽+𝛾},𝑔⟩_𝛽,𝛾 for all 𝑔∈𝐴^1𝛽 and 𝑓(𝑤)=⟨𝑓,𝐾^{𝑤𝛽+𝛾}⟩_𝛽,𝛾 for all 𝑓∈𝐻^∞𝛾.

Let 𝐸 be a Banach space of analytic functions on 𝔹 containing the constant functions. We denote its norm by ‖⋅‖_𝐸. With ‖⋅‖_𝐸^∗ we denote the norm of its Banach dual space 𝐸^∗. Consider the following conditions on 𝐸.(𝐶1)There are positive constants 𝑠 and 𝐶 such that ||𝑓(𝑧)||≤𝐶‖𝑓‖^𝐸1−|𝑧|²^𝑠,forevery𝑓∈𝐸,andforeach𝑧∈𝔹.(1.12)(𝐶2)The analytic polynomials are dense on 𝐸.(𝐶3)‖𝛿_𝑧‖_𝐸^∗→∞, when |𝑧|→1.(𝐶4)The linear span of the set {𝛿_𝑧∶𝑧∈𝔹} is dense in 𝐸^∗.(𝐶5)There is a 𝐶>0 such that ‖𝑓‖_𝐸≤𝐶‖𝑓‖_∞ for all 𝑓∈𝐸, where ‖𝑓‖_∞=sup_𝑧∈𝔹|𝑓(𝑧)|, the supremum taken in the extended real line.

It follows from (𝐶1) that the closed unit ball 𝐵_𝐸 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 (𝐶1) and (𝐶5). Then the contraction operators 𝐾_𝑟∶𝐸→𝐸 given by 𝐾_𝑟(𝑓)(𝑧)∶=𝑓(𝑟𝑧),0<𝑟<1,(1.13) are well defined and compact.

Proof. Each given 𝑓∈𝐸 may be approximated uniformly on compact subsets by the sequence (𝑆_𝑘)_𝑘∈ℕ0 of its Taylor polynomials at 0. Therefore, 𝑆_𝑘(𝑟𝑧)→𝑓(𝑟𝑧) uniformly on 𝔹 as 𝑘→∞. Further, (𝑆_𝑘(𝑟𝑧))_𝑘∈ℕ0⊂𝐸 is a Cauchy sequence by (𝐶5), so it converges to some element in 𝐸. Hence 𝐾_𝑟(𝑓)∈𝐸.
Moreover 𝐾_𝑟 is compact. Indeed, any sequence (𝑓_𝑚)_𝑚∈ℕ⊂𝐵_𝐸 has a 𝜏-convergent subsequence, say itself, to an element 𝑔∈𝐻(𝔹). Therefore (𝑓_𝑚(𝑟𝑧))_𝑚∈ℕ converges uniformly on 𝔹 to 𝑔(𝑟𝑧), and again (𝐶5) yields that (𝐾_𝑟(𝑓_𝑚))_𝑚∈ℕ is a Cauchy sequence in 𝐸 that converges to 𝐾_𝑟(𝑔).

We consider another condition on 𝐸.(𝐶6)For every 𝑓∈𝐸, we have ‖𝐾_𝑟(𝑓)‖_𝐸≤‖𝑓‖_𝐸 for 0<𝑟<1.

Remarks 1. (a) Note that (𝐶6) implies that ‖𝐾_𝑟‖_𝐸→𝐸=sup_{‖𝑓‖𝐸≤1}‖𝐾_𝑟(𝑓)‖_𝐸≤1.
(b) By (𝐶1) every functional at 𝑧∈𝔹_𝑛, 𝛿_𝑧∶𝐸→ℂ, is bounded, and therefore||𝛿_𝑧(𝑓)||≤‖‖𝛿_𝑧‖‖_𝐸∗‖𝑓‖_𝐸,forevery𝑓∈𝐸.(1.14)
 (c) Spaces like 𝐻^0𝑣, the little Bloch space, Hardy spaces 𝐻^𝑝 and weighted-Bergman spaces 𝐴^𝑝𝛽 fulfill conditions (𝐶1)–(𝐶6). For 𝐻^𝑝, 𝑠=𝑛/𝑝, and for 𝐴^𝑝𝛽, we have that 𝑠=(𝑛+𝛽+1)/𝑝 (see [9]).

Let 𝑋 and 𝑌 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+𝐾‖_𝑋→𝑌∶𝐾iscompact.(1.15)

We write 𝐴⪯𝐵 if there is a positive constant 𝐶, not depending on properties of 𝐴 and 𝐵, such that 𝐴≤𝐶𝐵. We also write 𝐴≃𝐵 whenever 𝐴⪯𝐵 and 𝐵⪯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 (𝐶1)–(𝐶6). Then there exists a sequence (𝐿_𝑚)_𝑚∈ℕ of compact operators on 𝐸 such that (i)for any 0<𝑡<1, lim_𝑚→∞sup_{‖𝑓‖𝐸≤1}sup_{|𝑧|≤𝑡}||𝐼−𝐿_𝑚(𝑓)(𝑧)||=0;(2.1)(ii)limsup_𝑚→∞‖‖𝐼−𝐿_𝑚‖‖_𝐸→𝐸≤1.(2.2)

Proof. We prove that for every 0<𝑡<1 and 𝜀>0 there is a compact operator 𝐿∶𝐸→𝐸 such that ‖𝐼−𝐿‖_𝐸→𝐸<1+2𝜀,(2.3) and sup_{‖𝑓‖𝐸≤1}sup_{|𝑧|≤𝑡}||(𝐼−𝐿)𝑓(𝑧)||<𝜀.(2.4) When this is done, a standard diagonal argument by taking a sequence (𝑡_𝑛)↑1 and a sequence of positive numbers (𝜀_𝑛)↓0 will give the result.
The operator 𝐿 will be constructed as a suitable finite convex combination of the operators 𝐾_𝑟 and therefore by Proposition 1.3, it will be compact.
The operators 𝐼−𝐾_𝑟∶(𝐻(𝔹),𝜏)→(𝐻(𝔹),𝜏) are continuous, and for each 𝑓∈𝐻(𝔹), we have that (𝐼−𝐾_𝑟)𝑓→0 in (𝐻(𝔹),𝜏) when 𝑟→1. By the Banach-Steinhaus theorem for Fréchet spaces (𝐼−𝐾_𝑟)→0 uniformly on relatively compact subsets of (𝐻(𝔹),𝜏) as 𝑟→1; thus in particular on 𝐵_𝐸. Hence, if we pick 𝑠₁ such that 𝑡<𝑠₁<1, we havelim_𝑟→1sup_{‖𝑓‖𝐸≤1}sup_{|𝑧|≤𝑠1}||𝐼−𝐾_𝑟𝑓(𝑧)||=0.(2.5)
Since 𝐵_𝐸 is an equicontinuous set, the operator 𝛿∶𝔹→𝐸^∗, 𝑧↦𝛿_𝑧 is continuous, so we can find 𝑧₀∈𝔹 such that sup_{|𝑧|≤𝑠1}‖𝛿_𝑧‖_𝐸^∗=‖𝛿_𝑧0‖_𝐸^∗. Moreover we have the inequalities1‖1‖_𝐸=||𝛿^𝑧(1)||‖1‖_𝐸≤‖‖𝛿_𝑧‖‖_𝐸∗.(2.6)
Therefore, we find an 𝑟₁∈(0,1) such thatsup_{‖𝑓‖𝐸≤1}sup_{|𝑧|≤𝑠1}||𝐼−𝐾_𝑟1𝑓(𝑧)||‖‖𝛿_𝑧‖‖_𝐸∗<min𝜀,𝜀‖‖𝛿_𝑧0‖‖_𝐸∗.(2.7)
Since the bounded set {𝛿_𝑧/‖𝛿_𝑧‖_𝐸^∗∶𝑧∈𝔹} is equicontinuous in 𝐸^∗, 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|→1‖‖𝛿_𝑧𝛿‖‖^𝑧_𝐸∗,𝑃=lim_|𝑧|→1𝑃(𝑧)‖‖𝛿_𝑧‖‖_𝐸∗=0,(2.8) since condition (𝐶3) holds and 𝑃 is bounded on 𝔹.
This means that lim_|𝑧|→1𝛿_𝑧/‖𝛿_𝑧‖_𝐸^∗=0 on relatively compact sets in 𝐸, in particular on 𝐾_𝑟1(𝐵_𝐸). Hence there is an 𝑠₂>𝑠₁ such that for each 𝑓∈𝐵_𝐸, we havesup_{|𝑧|≥𝑠2}||𝛿_𝑧𝐾_𝑟1𝑓||‖‖𝛿_𝑧‖‖_𝐸∗≤𝜀.(2.9) Therefore, sup_{‖𝑓‖𝐸≤1}sup_{|𝑧|≥𝑠2}||𝐼−𝐾_𝑟1𝑓(𝑧)||‖‖𝛿_𝑧‖‖_𝐸∗<1+𝜀.(2.10)
We can continue in this way and find two strictly increasing sequences (𝑠_𝑘) and (𝑟_𝑘) satisfyingsup_{‖𝑓‖𝐸≤1}sup_{|𝑧|≤𝑠𝑘}||𝐼−𝐾_𝑟𝑘𝑓(𝑧)||‖𝛿_𝑧‖_𝐸^∗<min𝜀,𝜀sup_{‖𝑓‖𝐸≤1}‖𝛿_𝑧0‖_𝐸^∗,(2.11)sup_{|𝑧|≥𝑠𝑘+1}||𝐼−𝐾_𝑟𝑘𝑓(𝑧)||‖𝛿_𝑧‖_𝐸^∗<1+𝜀.(2.12) Let 𝑚>1/𝜀 and put 𝐿=1𝑚^𝑚_𝑘=1𝐾_𝑟𝑘.(2.13) Then by (2.11) and the fact that ‖𝛿_𝑧0‖_𝐸^∗≥‖𝛿_𝑧‖_𝐸^∗ for |𝑧|≤𝑠₁, we have that (2.4) holds since sup_{‖𝑓‖𝐸≤1}sup_{|𝑧|≤𝑡}||(𝐼−𝐿)𝑓(𝑧)||≤1𝑚^𝑚_𝑘=1sup_{‖𝑓‖𝐸≤1}sup_{|𝑧|≤𝑠1}||𝐼−𝐾_𝑟𝑘𝑓(𝑧)||<𝜀.(2.14)
Now we show that (2.3) holds. Similarly to above, we have thatsup_{‖𝑓‖𝐸≤1}sup_|𝑧|<𝑡||(𝐼−𝐿)𝑓(𝑧)||‖𝛿_𝑧‖_𝐸^∗<𝜀.(2.15)
Moreover, by (2.12), if 𝑠_𝑙<|𝑧|≤𝑠_𝑙+1 and ‖𝑓‖_𝐸≤1, then||𝐼−𝐾_𝑟𝑘𝑓(𝑧)||‖‖𝛿_𝑧‖‖_𝐸∗<1+𝜀,(2.16) except possibly for 𝑘=𝑙, in which case ||𝐼−𝐾_𝑟𝑙𝑓(𝑧)||‖‖𝛿_𝑧‖‖_𝐸∗≤‖‖(𝐼−𝐾_𝑟𝑙)𝑓‖‖_𝐸≤‖‖𝐼−𝐾_𝑟𝑙‖‖_𝐸→𝐸≤2.(2.17) Hence, we get for all 𝑧∈𝔹 and ‖𝑓‖_𝐸≤1 that ||(𝐼−𝐿)𝑓(𝑧)||‖𝛿_𝑧‖_𝐸^∗≤1𝑚_𝑘≠𝑙||𝐼−𝐾_𝑟𝑘𝑓(𝑧)||‖𝛿_𝑧‖_𝐸^∗+1𝑚||𝐼−𝐾^𝑟𝑙𝑓(𝑧)||‖𝛿_𝑧‖_𝐸^∗<1+2𝜀.(2.18) In light of condition (𝐶4), we have ‖𝐼−𝐿‖_𝐸→𝐸=sup_𝑧∈𝔹‖‖‖(𝐼−𝐿)_∗𝛿_𝑧‖‖𝛿_𝑧‖‖_𝐸∗‖‖‖_𝐸∗=sup_𝑧∈𝔹sup_{‖𝑓‖𝐸≤1}||(𝐼−𝐿)𝑓(𝑧)||‖‖𝛿_𝑧‖‖_𝐸∗<1+2𝜀.(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 𝐻^∞𝑣 or 𝐻^0𝑣. We omit its proof.

Theorem 2.2. Suppose 𝑇∶𝐸→𝐻^∞𝑣 is a bounded operator. Then ‖𝑇‖_𝐸→𝐻^∞𝑣=sup_𝑧∈𝔹‖‖𝑇^∗(𝛿_𝑧)‖‖_𝐸∗‖‖𝛿_𝑧‖‖_𝑣.(2.20)

Since many operators can be written as weighted composition operators we state the following useful result. As usual, 𝑢𝐶_𝜑(𝑓)=𝑢⋅(𝑓∘𝜑) where 𝑢 is an analytic function on 𝔹 and 𝜑 is an analytic self-map of 𝔹.

Corollary 2.3. If 𝑢𝐶_𝜑∶𝐸→𝐻^∞𝑣 is a bounded weighted composition operator, then ‖‖𝑢𝐶_𝜑‖‖_{𝐸→𝐻∞𝑣}=sup_𝑧∈𝔹||𝑢(𝑧)||𝑣̃(𝑧)‖‖𝛿_𝜑(𝑧)‖‖_𝐸∗.(2.21)

Example 2.4. Let 𝐸=𝐴^𝑝𝛽, 𝛽>−1, 𝑝>1 with 1/𝑝+1/𝑞=1 and 𝑣_𝛼̃(𝑧)=𝑣_𝛼(𝑧)=(1−|𝑧|²)^𝛼, 𝛼>0. Then, since for all 𝑓∈𝐴^𝑝𝛽 and 𝑧∈𝔹, ||𝑓(𝑧)||≤‖𝑓‖^𝐴𝑝𝛽1−|𝑧|²^{(𝑛+1+𝛽)/𝑝},(2.22) and since equality is attained at 𝑓_𝑧(𝑤)=1−|𝑧|²^{(𝑛+1+𝛽)/𝑝}(1−⟨𝑤,𝑧⟩)^{2(𝑛+1+𝛽)/𝑝},(2.23) we get ‖‖𝛿_𝜑(𝑧)‖‖_𝐴𝑞𝛽=11−||𝜑(𝑧)||²^{(𝑛+1+𝛽)/𝑝}.(2.24) So we get ‖‖𝑢𝐶_𝜑‖‖_{𝐴𝑝𝛽→𝐻∞𝛼}=sup_𝑧∈𝔹1−|𝑧|²^𝛼||𝑢(𝑧)||1−||𝜑(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 (𝐶1)–(𝐶6). If 𝑇∶𝐸→𝐻^0𝑣 is a bounded operator, then ‖𝑇‖_𝑒=limsup_{|𝑧|→1−}‖‖𝑇^∗(𝛿_𝑧)‖‖_𝐸∗‖‖𝛿_𝑧‖‖_𝑣.(2.26)

Proof. If ℓ∶=limsup_{|𝑧|→1−}‖‖𝑇^∗(𝛿_𝑧)‖‖_𝐸∗‖‖𝛿_𝑧‖‖_𝑣,(2.27) then the same reasoning as in the first part of [3, Theorem 3.1] shows that ‖𝑇‖_𝑒≥ℓ.
To prove the reverse inequality, let (𝐿_𝑚)_𝑚∈ℕ be the sequence provided by Proposition 2.1. Then‖𝑇‖_𝑒≤‖‖𝑇−𝑇∘𝐿_𝑚‖‖_{𝐸→𝐻0𝑣},(2.28) for all 𝑚∈ℕ.
Next, we estimatelimsup_𝑚→∞‖‖𝑇−𝑇∘𝐿_𝑚‖‖_{𝐸→𝐻0𝑣}.(2.29)
Fix 0<𝑡<1 and 𝑓 in the unit ball of 𝐸. Then||𝑇∘𝐼−𝐿_𝑚(𝑓)(𝑧)||=||𝛿_𝑧,𝑇∘𝐼−𝐿_𝑚(𝑓)||=||𝐼−𝐿^∗𝑚∘𝑇^∗𝛿_𝑧,𝑓||.(2.30)
The sequence (𝐼−𝐿^∗𝑚)_𝑚∈ℕ⊂ℒ(𝐸^∗) is bounded because of (2.2) in Proposition 2.1. Since for each fixed 𝑧∈𝔹, and using (2.1) in Proposition 2.1lim_𝑚→∞‖‖(𝐼−𝐿^∗𝑚)(𝛿_𝑧)‖‖_𝐸∗=lim_𝑚→∞sup_{‖𝑓‖𝐸≤1}||𝐼−𝐿_𝑚(𝑓)(𝑧)||=0,(2.31) the sequence (𝐼−𝐿^∗𝑚)_𝑚∈ℕ converges to zero on every point in the total set {𝛿_𝑧∶𝑧∈𝔹}⊂𝐸^∗ as 𝑚→∞. So we appeal to the Banach-Steinhaus type theorem [15, III 4.5] to conclude that (𝐼−𝐿^∗𝑚)_𝑚∈ℕ converges to zero uniformly on compact subsets of 𝐸^∗. In particular on the image {𝑇^∗(𝛿_𝑧)∶|𝑧|≤𝑡}⊂𝐸^∗ of the compact set {𝛿_𝑧∶|𝑧|≤𝑡}⊂𝐺^∞𝑣, that is, 0=lim_𝑚→∞sup_{{|𝑧|≤𝑡}}‖‖𝐼−𝐿^∗𝑚𝑇^∗𝛿_𝑧‖‖_𝐸^∗=lim_𝑚→∞sup_{{‖𝑓‖𝐸≤1}}sup_{{|𝑧|≤𝑡}}||𝐼−𝐿^∗𝑚∘𝑇^∗𝛿_𝑧,𝑓||.(2.32) Therefore since the weight 𝑣̃ is bounded, lim_𝑚→∞sup_{{‖𝑓‖𝐸≤1}}sup_{{|𝑧|≤𝑡}}||𝑇∘𝐼−𝐿_𝑚(𝑓)(𝑧)||𝑣̃(𝑧)=0.(2.33) On the other hand, for |𝑧|>𝑡, and as in the proof of [3, Theorem 3.1] sup_{{‖𝑓‖𝐸≤1}}||𝑇∘𝐼−𝐿_𝑚(𝑓)(𝑧)||𝑣̃(𝑧)≤‖‖𝐼−𝐿_𝑚‖‖_{𝐸→𝐻0𝑣}⋅sup_|𝑧|>𝑡‖‖𝑇^∗(𝛿_𝑧)‖‖_𝐸∗‖‖𝛿_𝑧‖‖_𝑣.(2.34) Therefore, limsup_𝑚→∞‖‖𝑇−𝑇∘𝐿_𝑚‖‖_{𝐸→𝐻0𝑣}==lliimmssuup_𝑚→∞p_𝑚→∞sup_{{‖𝑓‖𝐸≤1}}sup_{|𝑧|<1}||𝑇∘𝐼−𝐿_𝑚(𝑓)(𝑧)||𝑣̃(𝑧)≤limsup_𝑚→∞sup_{{‖𝑓‖𝐸≤1}}sup_{{|𝑧|>𝑡}}||𝑇∘𝐼−𝐿_𝑚(𝑓)(𝑧)||𝑣̃(𝑧)‖𝐼−𝐿_𝑚‖_𝐸→𝐻^0𝑣⋅sup_|𝑧|>𝑡‖𝑇^∗𝛿_𝑧‖_𝐸^∗‖𝛿_𝑧‖_𝑣.(2.35) Applying Proposition 2.1 in (2.35) and letting 𝑡→1 we obtain ‖𝑇‖_𝑒≤ℓ.

Corollary 2.6. Assume that 𝐸 satisfies conditions (𝐶1)–(𝐶6). If 𝑢𝐶_𝜑∶𝐸→𝐻^0𝑣 is a bounded weighted composition operator, then ‖‖𝑢𝐶_𝜑‖‖_𝑒=limsup_{|𝑧|→1−}||𝑢(𝑧)||𝑣̃(𝑧)‖‖𝛿_𝜑(𝑧)‖‖_𝐸∗.(2.36)

Proof. We apply Theorem 2.5 and just use that (𝑢𝐶_𝜑)^∗(𝛿_𝑧)=𝑢(𝑧)𝛿_𝜑(𝑧) and that ‖𝛿_𝑧‖_𝑣=1/𝑣(̃𝑧).

Example 2.7. If 𝐸=𝐻^0𝑤, then from Corollary 2.6 we get the following extension of the one-dimensional result in [5, Theorem 2.2] ‖‖𝑢𝐶_𝜑‖‖_𝑒=limsup_{|𝑧|→1−}𝑣̃(𝑧)||𝑢(𝑧)||𝑤(𝜑(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 (𝐶1)–(𝐶6). Assume that 𝐸 has the 𝜆-metric approximation property or is a dual space. Let 𝑇∶𝐹→𝐻^∞𝑣 be a bounded operator such that 𝑇(𝐸)⊆𝐻^0𝑣 and whose restrictions to bounded subsets of 𝐹 are 𝜏-𝜏 continuous. Then ‖𝑇‖_𝑒≃limsup_{|𝑧|→1−}‖‖𝑇^∗(𝛿_𝑧)‖‖_𝐹∗‖‖𝛿_𝑧‖‖_𝑣.(2.38)

Proof. To see that 𝑇=(𝑇_|𝐸)^∗∗, it suffices to remark that 𝑇 is weak*-weak* continuous, that is, to check that 𝑙∘𝑇∈𝐸^∗ for all 𝑙∈𝐺^∞𝑣. And for this, it is enough to check that 𝑙∘𝑇 is weak* continuous on the closed unit ball 𝐵_𝐸^∗∗ of 𝐹=𝐸^∗∗, a fact that follows from observing that on (𝐵_𝐹,𝑤^∗) 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 𝜆-metric approximation property or is a dual space, we may use Axler et al. [16] to obtain that ‖𝑇‖_𝑒=‖(𝑇_|𝐸)^∗‖_𝑒≃‖𝑇_|𝐸‖_𝑒. From this and Theorem 2.5 the corollary follows.

Notice that spaces like 𝐻^∞𝑣, the Bloch space and 𝐵𝑀𝑂𝐴 satisfy the assumptions on 𝐹 in the above corollary.

Corollary 2.9. Let 𝛽>−1, 𝛾>0 and assume that 𝑇∶𝐴^1𝛽→𝐴^1𝛽 is a bounded operator. If 𝑇^(𝐻^0𝛾)⊆𝐻^0𝛾, where 𝑇^∶𝐻^∞𝛾→𝐻^∞𝛾 is the dual operator with respect to the duality ⟨⋅,⋅⟩_𝛽,𝛾, then ‖𝑇‖_𝑒≃limsup_{|𝑧|→1−}‖‖𝑇𝐾^{𝑧𝛽+𝛾}‖‖_𝐴1𝛽‖‖𝐾^{𝑧𝛽+𝛾}‖‖_𝐴1𝛽.(2.39)

Proof. Note that ‖𝐾^{𝑧𝛽+𝛾}‖_𝐴^1𝛽≃(1−|𝑧|²)^−𝛾. Let 𝑇^{|𝐻0𝛾} be the restriction to 𝐻^0𝛾 of 𝑇^. Then we can apply Theorem 2.5 so that ‖‖𝑇^|𝐻_0𝛾‖‖_𝑒=limsup_{|𝑧|→1−}‖‖‖𝑇_{|𝐻0𝛾}^∗(𝛿_𝑧)‖‖‖_{(𝐻0𝛾)∗}‖‖𝛿_𝑧‖‖_𝑣𝛾.(2.40) For ‖𝑓‖_𝐻^∞𝛾≤1, 𝑓∈𝐻^0𝛾 and 𝑧∈𝔻, |||𝑇_{|𝐻0𝛾}^∗𝛿_𝑧(𝑓)|||=||𝑇^𝑓(𝑧)||=|||𝐾^{𝑧𝛽+𝛾},𝑇^𝑓_𝛽,𝛾|||=|||𝑇𝐾_{𝑧𝛽+𝛾},𝑓_𝛽,𝛾|||.(2.41) Since by [12] 𝐻^0𝛾 is isomorphic to 𝑐₀, we obtain that ‖𝑇‖_𝑒≃‖‖(𝑇^{|𝐻0𝛾})^∗‖‖_𝑒≃‖‖𝑇^{|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 ℬ^𝑝/ℂ and ℬ^𝑝0/ℂ correspondingly. Similar subspaces of 𝐻^∞𝜇 and 𝐻^0𝜇 are denoted by 𝐻^∞𝜇/ℂ and 𝐻^0𝜇/ℂ correspondingly.

Observe that for any 𝑧∈𝔹, ‖𝛿_𝑧‖_(ℬ^𝑝0₎^∗≤1+‖𝛿_𝑧‖_(ℬ^𝑝0_/ℂ)^∗. Hence ℬ^𝑝0/ℂ being a subspace of ℬ^𝑝0 fulfills conditions (𝐶1)–(𝐶6).

Set 𝛿̂_𝑧(𝑓)∶=(1−|𝑧|²)^𝑝ℜ𝑓(𝑧) for 𝑧∈𝔹 and 𝑓∈ℬ_𝑝/ℂ.

Corollary 2.10. (a) Let 𝑇∶ℬ^𝑝0/ℂ→ℬ^𝑝0/ℂ be a bounded operator. Then ‖𝑇‖_𝑒=limsup_{|𝑧|→1−}‖‖𝑇^∗(𝛿̂_𝑧)‖‖_{(ℬ𝑝0/ℂ)∗}.(2.43)
 (b) Let 𝑇∶ℬ^𝑝/ℂ→ℬ^𝑝/ℂ be a bounded operator such that ℬ^𝑝0/ℂ is an invariant subspace for 𝑇 and whose restrictions to the bounded subsets are 𝜏-𝜏 continuous. Then ‖𝑇‖_𝑒≃limsup_{|𝑧|→1−}‖‖𝑇^∗(𝛿̂_𝑧)‖‖_{(ℬ𝑝/ℂ)∗}.(2.44)

Proof. Let ℜ∶ℬ^𝑝/ℂ→𝐻^∞𝑝/ℂ be defined by ℜ(𝑓)(𝑧)=ℜ𝑓(𝑧). Then both ℜ and ℜ∶ℬ^𝑝0/ℂ→𝐻^0𝑝/ℂ are onto isometries (see, e.g., [17]). Therefore, if we put 𝐿=ℜ∘𝑇, we have ‖𝐿‖_𝑒=‖𝑇‖_𝑒. Since ‖𝛿_𝑧‖_𝑣𝑝=(1−|𝑧|²)^−𝑝, and ℜ(𝑓)(𝑧)=ℜ^∗(𝛿_𝑧)(𝑓) we also have that 𝐿^∗‖‖𝛿𝛿_𝑧‖‖^𝑧_𝑣𝑝=𝑇^∗∘ℜ^∗‖‖𝛿𝛿_𝑧‖‖^𝑧_𝑣𝑝=𝑇^∗1−|𝑧|²^𝑝ℜ^∗𝛿_𝑧=𝑇^∗𝛿̂_𝑧.(2.45) Hence, ‖𝑇^∗(𝛿̂_𝑧)‖_(ℬ^𝑝0_/ℂ)^∗=‖𝐿^∗(𝛿_𝑧/‖𝛿_𝑧‖_𝑣)‖_(ℬ^𝑝0_/ℂ)^∗.
Now for (a), apply Theorem 2.5 to 𝐿. For (b), observe that 𝐿(ℬ^𝑝0/ℂ)⊆𝐻^0𝑝 and also that the quotient norm on ℬ^𝑝0/ℂ coincides with 𝑏_𝑝, and therefore using the quotient and the subspace duality, (ℬ^𝑝0/ℂ)^∗∗=ℬ^𝑝/ℂ. Moreover, 𝐿 is 𝜏-𝜏 continuous on bounded sets, because ℜ preserves the 𝜏-𝜏 continuity. Then use Corollary 2.8.

For a given 𝑔∈𝐻(𝔹), 𝑔(0)=0 and a holomorphic self-map 𝜑 of 𝔹, the following integral-type operator on 𝐻(𝔹) is introduced in [17] 𝑃^𝑔𝜑(𝑓)(𝑧)=¹⁰𝑓(𝜑(𝑡𝑧))𝑔(𝑡𝑧)𝑑𝑡𝑡,𝑓∈𝐻(𝔹),𝑧∈𝔹.(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] Stević calculated the essential norm of the operator 𝑃^𝑔𝜑∶𝐴^2𝛽→ℬ_𝜇. Motivated by this result we calculate here the essential norm of the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛽→ℬ_𝜇,0/ℂ, but in terms of the associated weight of 𝜇.

Corollary 2.11. Assume that 𝛽>−1, 𝑝>1, 𝑔∈𝐻(𝔹), 𝑔(0)=0, 𝜑 is a holomorphic self-map of 𝔹 and 𝑃^𝑔𝜑∶𝐴^𝑝𝛽→ℬ_𝜇,0/ℂ is bounded. Then ‖‖𝑃^𝑔𝜑‖‖_𝑒=limsup_|𝑧|→1𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛽)/𝑝}.(2.47)

Proof. Since ℜ𝑃^𝑔𝜑(𝑓)(𝑧)=𝑓(𝜑(𝑧))𝑔(𝑧), we have ℜ𝑃^𝑔𝜑=𝑔𝐶_𝜑∶𝐴^𝑝𝛽→𝐻^0𝜇/ℂ. From this and the fact that the operator ℜ is an onto isometry between ℬ_𝜇,0/ℂ and 𝐻^0𝜇/ℂ it follows that ‖𝑃^𝑔𝜑‖_𝑒=‖𝑔𝐶_𝜑‖_𝑒. By Corollary 2.6, we have ‖‖𝑔𝐶_𝜑‖‖_𝑒=limsup_{|𝑧|→1−}||𝑔(𝑧)||𝜇(𝑧)‖‖𝛿_𝜑(𝑧)‖‖_𝐴𝑞𝛽,(2.48) where 1/𝑝+1/𝑞=1. Therefore equality (2.47) follows simply bearing in mind (2.24).

Acknowledgments

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).