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

Authors: Pablo Galindo; Mikael Lindström; Stevo Stević
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The essential norm of any operator from a general Banach space
of holomorphic functions on the unit ball inℂn
into the little weighted-type
space is calculated. Some applications of the formula are given.


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## Body

## 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 Aα1(𝔻).We obtain a formula for the essential norm of any operator into a weighted-type space on the unit ball inℂn 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.Let𝔹 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,u∈𝔹 is denoted by ρ(z,u) (see [9] for more details). If X is a Banach space, by BX we denote the closed unit ball in X.Letv be a positive continuous function on 𝔹 (weight). The weighted-type space (or Bergman space of infinite order) Hv∞(𝔹)=Hv∞ is defined by (1.1)Hv∞={f∈H(B):‖f‖Hv∞:=sup⁡z∈Bv(z)|f(z)|<∞}.
The little weighted-type space Hv0(𝔹)=Hv0 consists of all f∈Hv∞ such that (1.2)lim⁡|z|→1v(z)|f(z)|=0.
With the norm ∥·∥Hv∞, both are Banach spaces. The norm topology of Hv∞ is finer than the topology τ of uniform convergence on compact subsets of 𝔹. For vα(z)=(1-|z|2)α, α>0, the standard weighted-type spaces, which we denote by Hα∞ and Hα0, are obtained. These spaces appear in the study of growth conditions of analytic functions, see, for example, [10, 11].The Bloch-type spaceℬμ(𝔹)=ℬμ consists of all f∈H(𝔹) such that (1.3)bμ(f)=sup⁡z∈Bμ(z)|Rf(z)|<∞,
where μ is a weight, (1.4)Rf(z)=∑j=1nzj∂f∂zj(z)=〈∇f(z),z¯〉
is the radial derivative of f, and ∇f is the complex gradient of f.The little Bloch-type spaceℬμ,0(𝔹)=ℬμ,0 consists of all f∈ℬμ such that (1.5)lim⁡|z|→1μ(z)|Rf(z)|=0.
Both are Banach spaces with the norm ∥f∥ℬμ=|f(0)|+bμ(f). For the standard weight vp, p>0, we get the p-Bloch space ℬp and the little p-Bloch space ℬ0p.The standard weighted-Bergman spaceAβp(𝔹)=Aβp, β>-1, p≥1, is the set of all analytic functions on 𝔹 such that (1.6)‖f‖Aβpp=∫B|f(z)|pcβ(1-|z|2)βdV(z)<∞,
where dV(z) is the normalized volume measure on 𝔹 and cβ=Γ(n+β+1)/n!Γ(β+1).A weightv 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 (1.7)ṽ(z):=(sup⁡{|f(z)|:f∈Hv∞,‖f‖Hv∞≤1})-1.
For a typical weight v the associated weight ṽ is also typical. Furthermore, for each z∈𝔹 there is an fz∈Hv∞, ∥fz∥Hv∞≤1, such that fz(z)=1/ṽ(z), and the same holds for the space Hv0. It is known that Hṽ∞≅Hv∞ and Hṽ0≅Hv0, that is, they are isometrically isometric [10].We say that a weightv satisfies condition L1 if it is radial and (L1)inf⁡k∈Nv(1-2-(k+1))v(1-2-k)>0.
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 Hα0 is isomorphic to c0.Since the closed unit ballBHv∞ is a compact subset of (H(𝔹),τ), a result of Dixmier-Ng [13] gives that the subspace of (Hv∞)*(1.8)Gv∞:={l∈(Hv∞)*:l∣BHv∞isτ-continuous}
is a predual of Hv∞, that is, (Gv∞)*≅Hv∞. Clearly the evaluation functional at z∈𝔹, defined by δz(f)=f(z), belongs to Gv∞. The norm of δz is denoted by ∥δz∥v. Moreover, the set {δz:z∈𝔹} is a total set, that is, its linear span is norm dense in Gv∞. More precisely, the next isomorphism result is due to Bierstedt and Summers.Lemma 1.1 (see [11]).
The mapf↦[l↦〈l,f〉] is an onto isometric isomorphism between Hv∞ and (Gv∞)* and the restriction map l↦l|Hv0 gives rise to an isometric isomorphism between Gv∞ and (Hv0)*.Similarly to the corresponding result in the one variable [14], one can prove the following.Lemma 1.2.
Supposeβ>-1 and γ>0. Then (Aβ1)* is isomorphic to Hγ0(1.9)〈f,g〉β,γ=∫Bf(z)g(z)¯cβ+γυβ+γ(z)dV(z),f∈Hγ∞,g∈Aβ1.
Moreover, under the pairing 〈f,g〉β,γ with g∈Hγ0 and f∈Aβ1, we also have that (Hγ0)* is isomorphic to Aβ1.Forz,w∈𝔹, let (1.10)Kzβ(w)=1(1-〈w,z〉)n+β+1.
Then the kernel function Kzβ+γ clearly belongs to Hγ0 and to Aβ1. It has the reproducing property (1.11)f(z)=∫Bf(w)Kzβ(w)¯cβυβ(w)dV(w),z∈B,
for every function f∈Aβ1 (see [9, Theorem 2.2]). As a direct application we get that g(w)=〈Kwβ+γ,g〉β,γ for all g∈Aβ1 and f(w)=〈f,Kwβ+γ〉β,γ for all f∈Hγ∞.LetE 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 constantss and C such that
(1.12)|f(z)|≤C‖f‖E(1-|z|2)s,foreveryf∈E,andforeachz∈B.(C2)
The analytic polynomials are dense onE.(C3)
∥δz∥E*→∞, when |z|→1.(C4)
The linear span of the set{δz:z∈𝔹} is dense in E*.(C5)
There is aC>0 such that ∥f∥E≤C∥f∥∞ for all f∈E, where ∥f∥∞=supz∈𝔹|f(z)|, the supremum taken in the extended real line.It follows from(C1) that the closed unit ball BE is τ-bounded, hence it is an equicontinuous and a τ-relatively compact set. Moreover, we have the following.Proposition 1.3.
Assume thatE satisfies conditions (C1) and (C5). Then the contraction operators Kr:E→E given by
(1.13)Kr(f)(z):=f(rz),0<r<1,
are well defined and compact.Proof.
Each givenf∈E may be approximated uniformly on compact subsets by the sequence (Sk)k∈ℕ0 of its Taylor polynomials at 0. Therefore, Sk(rz)→f(rz) uniformly on 𝔹 as k→∞. Further, (Sk(rz))k∈ℕ0⊂E is a Cauchy sequence by (C5), so it converges to some element in E. Hence Kr(f)∈E.
MoreoverKr is compact. Indeed, any sequence (fm)m∈ℕ⊂BE has a τ-convergent subsequence, say itself, to an element g∈H(𝔹). Therefore (fm(rz))m∈ℕ converges uniformly on 𝔹 to g(rz), and again (C5) yields that (Kr(fm))m∈ℕ is a Cauchy sequence in E that converges to Kr(g).We consider another condition onE.(C6)
For everyf∈E, we have ∥Kr(f)∥E≤∥f∥E for 0<r<1.Remarks 1.
(a) Note that(C6) implies that ∥Kr∥E→E=sup∥f∥E≤1∥Kr(f)∥E≤1.
(b) By (C1) every functional at z∈𝔹n, δz:E→ℂ, is bounded, and therefore(1.14)|δz(f)|≤‖δz‖E*‖f‖E,foreveryf∈E.
(c) Spaces likeHv0, the little Bloch space, Hardy spaces Hp and weighted-Bergman spaces Aβp fulfill conditions (C1)–(C6). For Hp, s=n/p, and for Aβp, we have that s=(n+β+1)/p (see [9]).LetX 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, (1.15)‖T‖e=inf⁡{‖T+K‖X→Y:Kiscompact}.We writeA⪯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 thatE satisfies conditions (C1)–(C6). Then there exists a sequence (Lm)m∈ℕ of compact operators on E such that (i)
for any0<t<1,
(2.1)lim⁡m→∞sup⁡‖f‖E≤1sup⁡|z|≤t|(I-Lm)(f)(z)|=0;(ii)
(2.2)limsup⁡m→∞‖I-Lm‖E→E≤1.Proof.
We prove that for every0<t<1 and ɛ>0 there is a compact operator L:E→E such that
(2.3)‖I-L‖E→E<1+2ɛ,
and
(2.4)sup⁡∥f∥E≤1sup⁡|z|≤t|(I-L)f(z)|<ɛ.
When this is done, a standard diagonal argument by taking a sequence (tn)↑1 and a sequence of positive numbers (ɛn)↓0 will give the result.
The operatorL will be constructed as a suitable finite convex combination of the operators Kr and therefore by Proposition 1.3, it will be compact.
The operatorsI-Kr:(H(𝔹),τ)→(H(𝔹),τ) are continuous, and for each f∈H(𝔹), we have that (I-Kr)f→0 in (H(𝔹),τ) when r→1. By the Banach-Steinhaus theorem for Fréchet spaces (I-Kr)→0 uniformly on relatively compact subsets of (H(𝔹),τ) as r→1; thus in particular on BE. Hence, if we pick s1 such that t<s1<1, we have(2.5)lim⁡r→1sup⁡‖f‖E≤1sup⁡|z|≤s1|(I-Kr)f(z)|=0.
SinceBE is an equicontinuous set, the operator δ:𝔹→E*, z↦δz is continuous, so we can find z0∈𝔹 such that sup|z|≤s1∥δz∥E*=∥δz0∥E*. Moreover we have the inequalities(2.6)1‖1‖E=|δz(1)|‖1‖E≤‖δz‖E*.
Therefore, we find anr1∈(0,1) such that(2.7)sup⁡‖f‖E≤1sup⁡|z|≤s1|(I-Kr1)f(z)|‖δz‖E*<min⁡{ɛ,ɛ‖δz0‖E*}.
Since the bounded set{δz/∥δz∥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,(2.8)lim⁡|z|→1〈δz‖δz‖E*,P〉=lim⁡|z|→1P(z)‖δz‖E*=0,
since condition (C3) holds and P is bounded on 𝔹.
This means thatlim|z|→1δz/∥δz∥E*=0 on relatively compact sets in E, in particular on Kr1(BE). Hence there is an s2>s1 such that for each f∈BE, we have(2.9)sup⁡|z|≥s2|δz(Kr1f)|‖δz‖E*≤ɛ.
Therefore,
(2.10)sup⁡‖f‖E≤1sup⁡|z|≥s2|(I-Kr1)f(z)|‖δz‖E*<1+ɛ.
We can continue in this way and find two strictly increasing sequences(sk) and (rk) satisfying(2.11)sup⁡∥f∥E≤1sup⁡|z|≤sk|(I-Krk)f(z)|∥δz∥E*<min⁡{ɛ,ɛ∥δz0∥E*},(2.12)sup⁡∥f∥E≤1sup⁡|z|≥sk+1|(I-Krk)f(z)|∥δz∥E*<1+ɛ.
Let m>1/ɛ and put
(2.13)L=1m∑k=1mKrk.
Then by (2.11) and the fact that ∥δz0∥E*≥∥δz∥E* for |z|≤s1, we have that (2.4) holds since
(2.14)sup⁡∥f∥E≤1sup⁡|z|≤t|(I-L)f(z)|≤1m∑k=1msup⁡∥f∥E≤1sup⁡|z|≤s1|(I-Krk)f(z)|<ɛ.
Now we show that (2.3) holds. Similarly to above, we have that(2.15)sup⁡∥f∥E≤1sup⁡|z|<t|(I-L)f(z)|∥δz∥E*<ɛ.
Moreover, by (2.12), if sl<|z|≤sl+1 and ∥f∥E≤1, then(2.16)|(I-Krk)f(z)|‖δz‖E*<1+ɛ,
except possibly for k=l, in which case
(2.17)|(I-Krl)f(z)|‖δz‖E*≤‖(I-Krl)f‖E≤‖I-Krl‖E→E≤2.
Hence, we get for all z∈𝔹 and ∥f∥E≤1 that
(2.18)|(I-L)f(z)|∥δz∥E*≤1m∑k≠l|(I-Krk)f(z)|∥δz∥E*+1m|(I-Krl)f(z)|∥δz∥E*<1+2ɛ.
In light of condition (C4), we have
(2.19)‖I-L‖E→E=sup⁡z∈B‖(I-L)*δz‖δz‖E*‖E*=sup⁡z∈Bsup⁡∥f∥E≤1|(I-L)f(z)|‖δz‖E*<1+2ε.
Thus we also have (2.3), and the statement is proved.Next we state a formula for the norm of any operator fromE to either Hv∞ or Hv0. We omit its proof.Theorem 2.2.
SupposeT:E→Hv∞ is a bounded operator. Then
(2.20)‖T‖E→Hv∞=sup⁡z∈B‖T*(δz)‖E*‖δz‖v.Since many operators can be written as weighted composition operators we state the following useful result. As usual,uCφ(f)=u·(f∘φ) where u is an analytic function on 𝔹 and φ is an analytic self-map of 𝔹.Corollary 2.3.
IfuCφ:E→Hv∞ is a bounded weighted composition operator, then
(2.21)‖uCφ‖E→Hv∞=sup⁡z∈B|u(z)|ṽ(z)‖δφ(z)‖E*.Example 2.4.
LetE=Aβp, β>-1, p>1 with 1/p+1/q=1 and ṽα(z)=vα(z)=(1-|z|2)α, α>0. Then, since for all f∈Aβp and z∈𝔹,
(2.22)|f(z)|≤‖f‖Aβp(1-|z|2)(n+1+β)/p,
and since equality is attained at
(2.23)fz(w)=(1-|z|2)(n+1+β)/p(1-〈w,z〉)2(n+1+β)/p,
we get
(2.24)‖δφ(z)‖Aβq=1(1-|φ(z)|2)(n+1+β)/p.
So we get
(2.25)‖uCφ‖Aβp→Hα∞=sup⁡z∈B(1-|z|2)α|u(z)|(1-|φ(z)|2)(n+1+β)/p.Now we are ready for our main result. Compare it with Theorem2.2.Theorem 2.5.
Assume thatE satisfies conditions (C1)–(C6). If T:E→Hv0 is a bounded operator, then
(2.26)‖T‖e=limsup⁡|z|→1-‖T*(δz)‖E*‖δz‖v.Proof.
If(2.27)l:=limsup⁡|z|→1-‖T*(δz)‖E*‖δz‖v,
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(Lm)m∈ℕ be the sequence provided by Proposition 2.1. Then(2.28)‖T‖e≤‖T-T∘Lm‖E→Hv0,
for all m∈ℕ.
Next, we estimate(2.29)limsup⁡m→∞‖T-T∘Lm‖E→Hv0.
Fix0<t<1 and f in the unit ball of E. Then(2.30)|[T∘(I-Lm)](f)(z)|=|〈δz,[T∘(I-Lm)](f)〉|=|〈(I-Lm*)∘T*(δz),f〉|.
The sequence(I-Lm*)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 2.1(2.31)lim⁡m→∞‖(I-Lm*)(δz)‖E*=lim⁡m→∞sup⁡∥f∥E≤1|(I-Lm)(f)(z)|=0,
the sequence (I-Lm*)m∈ℕ converges to zero on every point in the total set {δz:z∈𝔹}⊂E* as m→∞. So we appeal to the Banach-Steinhaus type theorem [15, III 4.5] to conclude that (I-Lm*)m∈ℕ converges to zero uniformly on compact subsets of E*. In particular on the image {T*(δz):|z|≤t}⊂E* of the compact set {δz:|z|≤t}⊂Gv∞, that is,
(2.32)0=lim⁡m→∞sup⁡{|z|≤t}‖(I-Lm*)(T*(δz))‖E*=lim⁡m→∞sup⁡{∥f∥E≤1}sup⁡{|z|≤t}|〈(I-Lm*)∘T*(δz),f〉|.
Therefore since the weight ṽ is bounded,
(2.33)lim⁡m→∞sup⁡{∥f∥E≤1}sup⁡{|z|≤t}|[T∘(I-Lm)](f)(z)|ṽ(z)=0.
On the other hand, for |z|>t, and as in the proof of [3, Theorem 3.1]
(2.34)sup⁡{∥f∥E≤1}|[T∘(I-Lm)](f)(z)|ṽ(z)≤‖I-Lm‖E→Hv0⋅sup⁡|z|>t‖T*(δz)‖E*‖δz‖v.
Therefore,
(2.35)limsup⁡m→∞‖T-T∘Lm‖E→Hv0=limsup⁡m→∞sup⁡{∥f∥E≤1}sup⁡{|z|<1}|[T∘(I-Lm)](f)(z)|ṽ(z)=limsup⁡m→∞sup⁡{∥f∥E≤1}sup⁡{|z|>t}|[T∘(I-Lm)](f)(z)|ṽ(z)≤limsup⁡m→∞∥I-Lm∥E→Hv0⋅sup⁡|z|>t∥T*(δz)∥E*∥δz∥v.
Applying Proposition 2.1 in (2.35) and letting t→1 we obtain ∥T∥e≤ℓ.Corollary 2.6.
Assume thatE satisfies conditions (C1)–(C6). If uCφ:E→Hv0 is a bounded weighted composition operator, then
(2.36)‖uCφ‖e=limsup⁡|z|→1-|u(z)|ṽ(z)‖δφ(z)‖E*.Proof.
We apply Theorem2.5 and just use that (uCφ)*(δz)=u(z)δφ(z) and that ∥δz∥v=1/ṽ(z).Example 2.7.
IfE=Hw0, then from Corollary 2.6 we get the following extension of the one-dimensional result in [5, Theorem 2.2]
(2.37)‖uCφ‖e=limsup⁡|z|→1-ṽ(z)|u(z)|w̃(φ(z)).Corollary 2.8.
LetF 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→Hv∞ be a bounded operator such that T(E)⊆Hv0 and whose restrictions to bounded subsets of F are τ-τ continuous. Then
(2.38)‖T‖e≃limsup⁡|z|→1-‖T*(δz)‖F*‖δz‖v.Proof.
To see thatT=(T|E)**, it suffices to remark that T is weak*-weak* continuous, that is, to check that l∘T∈E* for all l∈Gv∞. And for this, it is enough to check that l∘T is weak* continuous on the closed unit ball BE** of F=E**, a fact that follows from observing that on (BF,w*) the weak*-topology coincides with the Hausdorff coarser one τ, for which both T and l are also continuous on bounded sets.
SinceE 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 likeHv∞, the Bloch space and BMOA satisfy the assumptions on F in the above corollary.Corollary 2.9.
Letβ>-1, γ>0 and assume that T:Aβ1→Aβ1 is a bounded operator. If T′(Hγ0)⊆Hγ0, where T′:Hγ∞→Hγ∞ is the dual operator with respect to the duality 〈·,·〉β,γ, then
(2.39)‖T‖e≃limsup⁡|z|→1-‖T(Kzβ+γ)‖Aβ1‖Kzβ+γ‖Aβ1.Proof.
Note that∥Kzβ+γ∥Aβ1≃(1-|z|2)-γ. Let T′|Hγ0 be the restriction to Hγ0 of T′. Then we can apply Theorem 2.5 so that
(2.40)‖T′|Hγ0‖e=limsup⁡|z|→1-‖(T′|Hγ0)*(δz)‖(Hγ0)*‖δz‖vγ.
For ∥f∥Hγ∞≤1, f∈Hγ0 and z∈𝔻,
(2.41)|(T′|Hγ0)*δz(f)|=|T′f(z)|=|〈Kzβ+γ,T′f〉β,γ|=|〈T(Kzβ+γ),f〉β,γ|.
Since by [12] Hγ0 is isomorphic to c0, we obtain that
(2.42)‖T‖e≃‖(T′|Hγ0)*‖e≃‖T′|Hγ0‖e,
as wanted.Now we apply Theorem2.5 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/ℂ and ℬ0p/ℂ correspondingly. Similar subspaces of Hμ∞ and Hμ0 are denoted by Hμ∞/ℂ and Hμ0/ℂ correspondingly.Observe that for anyz∈𝔹, ∥δz∥(ℬ0p)*≤1+∥δz∥(ℬ0p/ℂ)*. Hence ℬ0p/ℂ being a subspace of ℬ0p fulfills conditions (C1)–(C6).Setδ̂z(f):=(1-|z|2)pℜf(z) for z∈𝔹 and f∈ℬp/ℂ.Corollary 2.10.
(a) LetT:ℬ0p/ℂ→ℬ0p/ℂ be a bounded operator. Then
(2.43)‖T‖e=limsup⁡|z|→1-‖T*(δ̂z)‖(B0p/C)*.
(b) LetT:ℬp/ℂ→ℬp/ℂ be a bounded operator such that ℬ0p/ℂ is an invariant subspace for T and whose restrictions to the bounded subsets are τ-τ continuous. Then(2.44)‖T‖e≃limsup⁡|z|→1-‖T*(δ̂z)‖(Bp/C)*.Proof.
Letℜ:ℬp/ℂ→Hp∞/ℂ be defined by ℜ(f)(z)=ℜf(z). Then both ℜ and ℜ:ℬ0p/ℂ→Hp0/ℂ are onto isometries (see, e.g., [17]). Therefore, if we put L=ℜ∘T, we have ∥L∥e=∥T∥e. Since ∥δz∥vp=(1-|z|2)-p, and ℜ(f)(z)=ℜ*(δz)(f) we also have that
(2.45)L*(δz‖δz‖vp)=T*∘R*(δz‖δz‖vp)=T*((1-|z|2)pR*(δz))=T*(δ̂z).
Hence, ∥T*(δ̂z)∥(ℬ0p/ℂ)*=∥L*(δz/∥δz∥v)∥(ℬ0p/ℂ)*.
Now for (a), apply Theorem2.5 to L. For (b), observe that L(ℬ0p/ℂ)⊆Hp0 and also that the quotient norm on ℬ0p/ℂ coincides with bp, and therefore using the quotient and the subspace duality, (ℬ0p/ℂ)**=ℬp/ℂ. Moreover, L is τ-τ continuous on bounded sets, because ℜ preserves the τ-τ continuity. Then use Corollary 2.8.For a giveng∈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] (2.46)Pφg(f)(z)=∫01f(φ(tz))g(tz)dtt,f∈H(B),z∈B.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 Pφg:Aβ2→ℬμ. Motivated by this result we calculate here the essential norm of the operator Pφg:Aβp→ℬμ,0/ℂ, but in terms of the associated weight of μ.Corollary 2.11.
Assume thatβ>-1, p>1, g∈H(𝔹), g(0)=0, φ is a holomorphic self-map of 𝔹 and Pφg:Aβp→ℬμ,0/ℂ is bounded. Then
(2.47)‖Pφg‖e=limsup⁡|z|→1μ̃(z)|g(z)|(1-|φ(z)|2)(n+1+β)/p.Proof.
SinceℜPφg(f)(z)=f(φ(z))g(z), we have ℜPφg=gCφ:Aβp→Hμ0/ℂ. From this and the fact that the operator ℜ is an onto isometry between ℬμ,0/ℂ and Hμ0/ℂ it follows that ∥Pφg∥e=∥gCφ∥e. By Corollary 2.6, we have
(2.48)‖gCφ‖e=limsup⁡|z|→1-|g(z)|μ̃(z)‖δφ(z)‖Aβq,
where 1/p+1/q=1. Therefore equality (2.47) follows simply bearing in mind (2.24).



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*Source: 939873-2011-07-24.xml*