# A New Essential Norm Estimate of Composition Operators from Weighted Bloch Space intoμ-Bloch Spaces

Authors: René E. Castillo; Julio C. Ramos-Fernández; Edixon M. Rojas
Journal: Journal of Function Spaces and Applications
 (2013)

---

## Abstract

Letμ be any weight function defined on the unit disk 𝔻 and let ϕ be an analytic self-map of 𝔻. In the present paper, we show that the essential norm of composition operator Cϕ mapping from the weighted Bloch space to μ-Bloch space ℬμ is comparable to limsup|a|→1-∥σa∘ϕ∥ℬμ, where for a∈𝔻,  σa is a certain special function in the weighted Bloch space. As a consequence of our estimate, we extend the results about the compactness of composition operators due to Tjani (2003).


---

## Body

## 1. Introduction

The weighted Bloch space appears in the literature when we study properties of certain operators acting on certain spaces of analytic functions on the unit disk𝔻 of the complex plane ℂ. For instance, in 1991, Brown and Shields [1] showed that an analytic function u is a multiplier on the Bloch space ℬ if and only if
(1)∥u∥*=supz∈D(1-|z|2)log(11-|z|2)|u′(z)|<∞.
Also, in 1992, Attele [2] showed that Hankel operator induced by a function f∈H(𝔻) in the Bergman space La1 (for the definition of Bergman space and the Hankel operator, see [3]) is bounded if and only if ∥f∥*<∞, where H(𝔻)is the space of all holomorphic functions on 𝔻 with the topology of uniform convergence on compact subsets of 𝔻. The set of all functions f∈H(𝔻) such that (1) holds is denoted by ℬlog and it is known that ℬlog is a Banach space with the norm ∥f∥ℬlog=|f(0)|+∥f∥*.In the last decade, many authors have studied different classes of Bloch-type spaces, where the weight function,v(z)=(1-|z|2)log(1/(1-|z|2)),  (z∈𝔻), is replaced by a bounded, continuous, and positive function μ defined on 𝔻. More precisely, a function f∈H(𝔻) is called a μ-Bloch function, denoted as f∈ℬμ, if
(2)∥f∥μ:=supz∈𝔻μ(z)|f′(z)|<∞.
If μ1(z)=1-|z|2, we get the classical Bloch space ℬ (see [3]), while if μ2(z)=μ1α(z) with α>0, ℬμis just the α-Bloch space (see [4]). It is readily seen that Bμ is a Banach space with the norm ∥f∥ℬμ:=|f(0)|+∥f∥μ.A holomorphic functionϕ from the unit disk 𝔻 into itself induces a linear operator Cϕ, defined by Cϕ(f)=f∘ϕ, where f∈H(𝔻). Cϕ is the so-called composition operator with symbolϕ.Composition operators continue to be widely studied on many subspaces of H(𝔻)and particularly in Bloch-type spaces.The study of the properties of composition operators on Bloch-type spaces began with the celebrated work of Madigan and Matheson in [5], where they characterized the continuity and compactness of composition operators acting on the Bloch space ℬ. Many extensions of Madigan and Matheson’s results have appeared (see, e.g., [6] and a lot of references therein). An important reference on this subject is the work of Montes-Rodríguez in [7] (see also [8]); here the author supposes that weight μ is radial (μ(|z|)=μ(z) for all z∈𝔻), typical (lim|z|→1-μ(z)=0), and decreasing function of |z| defined on 𝔻. In particular, Yoneda in [9] has extended the results by Madigan and Matheson in [5] to composition operators Cϕ acting between weighted Bloch spaces. The weight considered by Yoneda in [9] was μlog(z)=(1-|z|2)log(2/(1-|z|2)) which is not decreasing on 𝔻.Recently, many authors have found new criteria for the continuity and compactness of composition operators acting on Bloch-type spaces in terms of thenth power of the symbol ϕ and the norm of the nth power of the identity function on 𝔻. The first result of this kind appears in 2009 and it is due to Wulan et al. [10]; in turn, their result was extended to α-Bloch spaces by Zhao in [11]. Another criterion for the continuity and compactness of composition operators on Bloch space is due to Tjani in [12] (see also [13] or more recently [10]); she showed the following result.Theorem 1 (see [12]).
The composition operatorCϕ is compact on ℬ if and only if  ϕ∈ℬ and
(3)lim|a|→1-∥φa∘ϕ∥ℬ=0,
where φa is a Möbius transformation from the unit disk onto itself; that is, φa(z)=(a-z)/(1-a¯z), with z∈𝔻.This last result has been recently extended toα-Bloch spaces by Malavé-Ramírez and Ramos-Fernández in [14].The essential norm of a continuous linear operatorT between normed linear spaces X and Y is its distance from the compact operators; that is, ∥T∥eX→Y=inf{∥T-K∥X→Y:K:X→Yiscompact}, where ∥·∥X→Y denotes the operator norm. Notice that ∥T∥eX→Y=0 if and only if T is compact, so that estimates on ∥T∥eX→Y lead to conditions for T to be compact. The essential norm of a composition operator on ℬ was calculated by Montes-Rodríguez in [15]. He obtained similar results for essential norms of weighted composition operators between weighted Banach spaces of analytic functions in [7]. Recently, many extensions of the above results have appeared in the literature; for instance, Zhao in [11] gave a formula for the essential norm of Cϕ:ℬα→ℬβ in terms of an expression involving norms of powers of ϕ. More precisely, he showed that
(4)∥Cϕ∥eℬα→ℬβ=(e2α)αlimsupj→∞jα-1∥ϕj∥ℬβ.
It follows from the discussion at the beginning of this paragraph that Cϕ:ℬα→ℬβ is compact if and only if
(5)limj→∞jα-1∥ϕj∥ℬβ=0.
Zhao’s results in [11] have been extended recently to the weighted Bloch spaces by Castillo et  al. in [16]. Also, Hyvärinen et al. in [17] obtained necessary and sufficient conditions for boundedness and an expression characterizing the essential norm of a weighted composition operator between general weighted Bloch spaces ℬμ, under the technical requirements that μ is radial, and that it is nonincreasing and tends to zero toward the boundary of 𝔻.The goal of the present paper is to give a new estimate of the essential norm of compositionCϕ mapping from ℬlog to ℬμ. More precisely, in the next section we will show the following result.Theorem 2.
Letϕ be an analytic self-map of the unit disk 𝔻. Then for the essential norm of the composition operator Cϕ:ℬlog→ℬμ, one has
(6)∥Cϕ∥eℬlog→ℬμ:limsup|a|→1-∥σa∘ϕ∥ℬμ.The relation (6) means that there is a positive constant M such that
(7)1M∥Cϕ∥eℬlog→ℬμ≤limsup|a|→1-∥σa∘ϕ∥ℬμ≤M∥Cϕ∥eℬlog→ℬμ,
and the functions σa with a∈𝔻 will be defined at the beginning of the next section. As a consequence of our estimate, we extend recent results, about the compactness of composition operators, due to Tjani in [12] (see Theorem 1) and Malavé-Ramírez and Ramos-Fernández in [14].


## 2. A Class of Special Functions forℬlog

For convenience, throughout this paper we consider the following weight:(8)vlog(z)=(1-|z|2)log(e1-|z|2),(z∈𝔻),
in place of the one used by Brown and Shields in (1). It is not hard to see that with this new weight we have the same space ℬlog, and the seminorm
(9)∥f∥log:=supz∈𝔻vlog(z)|f′(z)|
is comparable to ∥f∥* defined in (1). Moreover, this weight vlog satisfies the following property which will be of main relevance in the proof of our result.Lemma 3.
For allr∈(0,1) and all z∈𝔻, one has
(10)rvlog(z)≤vlog(rz).Proof.
The result follows from the fact that the functionh(t)=tlog(e/t) with t∈(0,1) is concave and satisfies h(t)→0 as t→0+.The key to our results lies in considering the following family of functions: fora∈𝔻 fixed, we define
(11)σa(z)=(1-|a|)(H(a¯z)-1),(z∈𝔻),
where for a∈𝔻, the function H is defined by
(12)H(z)=1(1-z)log(e/(1-z)).
We have the following result.Lemma 4.
The family{σa}a∈𝔻 satisfies the following properties. (1)
σa converges to zero uniformly on compact subsets of 𝔻 as |a|→1-.(2)
There exists a constantCσ>0 such that
(13)vlog(a)|σa′(a)|≥Cσ
for all a∈𝔻 such that (1/2)<|a|<1.(3)
There exists a constantMσ>0 such that
(14)supa∈𝔻∥σa∥ℬlog≤Mσ.Proof.
The property (1) is clear since the functionH is bounded on compact subsets of 𝔻.
(2) If(1/2)<|a|<1, then
(15)|σa′(a)|=|a|(1-|a|)log(1-|a|2)(1-|a|2)2log2(e/(1-|a|2))(16)≥12log(1-|a|2)1-log(1-|a|2)m×1(1-|a|2)log(1-|a|2),
and the property follows since
(17)limt→0+log(t)1-log(t)=1.
(3) Indeed, fora∈𝔻, we have
(18)∥σa∥ℬlog=supz∈𝔻|a|(1-|z|2)log(e/(1-|z|2))(1-|a|)|log(1-a¯z)||(1-a¯z)log(e/(1-a¯z))|2≤supz∈𝔻(1-|z|2)log(e/(1-|z|2))(1-|a|)|log(1-a¯z)||1-a¯z|2log2(e/|1-a¯z|)
since, for each w∈ℂ, we have |log(w)|≥log|w|. Note that the last expression is uniformly bounded if 1≤|1-a¯z|<2, and thus it will be enough to consider the case when |1-a¯z|<1. Furthermore, since the function h(t)=tlog(e/t) with t∈(0,1) is increasing, concave and satisfy h(t)→0 as t→0+, then we can use the fact that 1>|1-a¯z|≥(1/2)(1-|z|2) and conclude that
(19)h(|1-a¯z|)≥h(12(1-|z|2))≥12h(1-|z|2).
Hence
(20)(1-|z|2)log(e/(1-|z|2))|1-a¯z|log(e/|1-a¯z|)≤2.
Also, it is clear that
(21)1-|a||1-a¯z|≤1.
Finally, since
(22)limt→0+log2t+4π21-logt=1,
then we can deduce that there exists a constant M>0 such
(23)|log(1-a¯z)|log(e/|1-a¯z|)≤log2|1-a¯z|+4π21-log|1-a¯z|≤M.
Now the proof is complete.


## 3. A Criterion for the Compactness of Operators Acting on Spaces of Analytic Functions

We will need to prove three auxiliary facts. The first of which is the following lemma that appears in [18] and also in [12]. In both of these places, we point out a typographical error (“point evaluation functionals on X" there, as one can see from Relation (16) in [12], should be “point evaluations on Y").Lemma 5 (Tjani).
LetX, Y be two Banach spaces of analytic functions on 𝔻. Suppose that (1)
point evaluation functionals onY are bounded;(2)
the closed unit ball ofX is a compact subset of X in the topology of uniform convergence on compact sets;(3)
T:X→Y is continuous when X and Y are equipped with the topology of uniform convergence on compact sets.
ThenT is a compact operator if and only if given a bounded sequence (fj)j∈ℕ in X such that fj→0 uniformly on compact sets, (Tfj)j∈ℕ→0 in Y.The following facts aboutμ-Bloch space are well known. (i)
For each compactK⊂𝔻, there is a CK≥0 such that for all f∈ℬμ and z∈K, we have that
(24)|f(z)|≤CK∥f∥ℬμ.(ii)
Every point evaluation functional onℬμ is bounded.(iii)
The closed unit ball ofℬμ is compact in the topology of uniform convergence on compact subsets of 𝔻.(iv)
Ifγ is a weight on 𝔻 and ϕ:𝔻→𝔻 is holomorphic, with Cϕ:ℬμ→ℬγ, then Cϕ is continuous with respect to the compact-open subspace topologies on ℬμ and ℬγ.Thus, combining the above and Lemma5, we obtain the following auxiliary result.Lemma 6.
Letμ1 and μ2 be weights on 𝔻 and suppose that ϕ:𝔻→𝔻 is holomorphic. Then Cϕ:ℬμ1→ℬμ2 is compact if and only if given a bounded sequence (fj)j∈N in ℬμ1 such that fj→0uniformly on compact subsets of 𝔻, then ∥Cϕ(fj)∥ℬμ2→0 as j→∞.As a consequence of Lemma5, we can see that the dilatation operator is compact on ℬlog. Recall that for r∈[0,1], the linear dilation operatorKr:H(𝔻)→H(𝔻) is defined by Krf=fr, where fr, for each f∈H(𝔻), is given by fr(z)=f(rz). It is clear that if f∈H(𝔻) then rfr→f uniformly on compact subsets of 𝔻 as r→1-.Lemma 7.
The following statements hold:(1)
forr∈[0,1), the operator Kr is compact on ℬlog;(2)
for eachr∈[0,1](25)∥Kr∥ℬlog→ℬlog≤1.Proof.
If{fn} is a bounded sequence in ℬlog such that fn→0 uniformly on compact subsets of 𝔻, then
(26)0≤limn→∞∥Krfn∥log=supz∈𝔻vlog(z)r|f′(rz)|≤∥vlog∥∞limn→∞sups∈D¯(0,r)|f′(s)|=0,
and the item (1) follows by Lemma 5.
On the other hand, iff∈ℬlog then
(27)∥Krf∥ℬlog=|f(0)|+supz∈𝔻rvlog(z)vlog(rz)vlog(rz)|f′(rz)|≤|f(0)|+∥f∥log=∥f∥ℬlog,
where we have used Lemma 3. This ends the proof of item (2).


## 4. Proof of Theorem2

Now we can show Theorem2. We set
(28)L=limsup|a|→1-∥σa∘ϕ∥ℬμ.
Let K:ℬlog→ℬμ be any compact operator, a∈𝔻 fixed and define
(29)fa(z)=1Mσσa(z),(z∈𝔻),
where Mσ is the constant in the item (3) of Lemma 4. Then fa goes to zero uniformly on compact subsets of 𝔻 as |a|→1-, ∥fa∥ℬlog≤1 for all a∈𝔻, and
(30)∥Cϕ-K∥ℬlog→ℬμ≥∥(Cϕ-K)fa∥ℬμ≥1Mσ∥σa∘ϕ∥ℬμ-∥Kfa∥ℬμ.
Hence, taking limsup|a|→1- and using Lemma 5, we obtain
(31)∥Cϕ∥eℬlog→ℬμ≥1Mσlimsup|a|→1-∥σa∘ϕ∥ℬμ.Now, we go to show that there exists a constantC>0 such that
(32)∥Cϕ∥eℬlog→ℬμ≤Climsup|a|→1-∥σa∘ϕ∥ℬμ.Hence, if we consider a sequence{rn}⊂(0,1) such that rn→1 as n→∞ and we consider the dilatation operators Kn=Krn, then by Lemma 7, for all n∈ℕ, the operator CϕKn is a compact from ℬlog into ℬμ, and by definition of the essential norm, we have
(33)∥Cϕ∥eℬlog→ℬμ≤limsupn→∞∥Cϕ-CϕKn∥ℬlog→ℬμ.
Thus, we have to show that there exists a constant M>0 such that
(34)limsupn→∞∥Cϕ-CϕKn∥ℬlog→ℬμ≤ML.
To see this, consider any f∈ℬlog such that ∥f∥ℬlog=1; then since
(35)∥(Cϕ-CϕKn)f∥ℬμ=|f(ϕ(0))-f(rnϕ(0))|+∥(f-frn)∘ϕ∥μ
and |f(ϕ(0))-f(rnϕ(0))|→0 as n→∞, it is enough to show that there exists a constant M>0 such that
(36)limsupn→∞∥(f-frn)∘ϕ∥μ≤ML.
Furthermore, since rn(f′)rn→f′ uniformly on compact subsets of 𝔻 as n→∞, we have
(37)limsupn→∞sup|ϕ(z)|≤rNμ(z)|(f-frn)′(ϕ(z))||ϕ′(z)|=0,
where N∈ℕ is large enough such that rn≥(1/2) for all n≥N. Hence we only have to show that there exists a constant M>0 such that
(38)S≔limsupn→∞sup|ϕ(z)|>rNμ(z)|(f-frn)′(ϕ(z))||ϕ′(z)|≤ML.
Indeed, we write S≤limsupn→∞(S1+S2), where
(39)S1=sup|ϕ(z)|>rNμ(z)|f′(ϕ(z))||ϕ′(z)|,S2=sup|ϕ(z)|>rNμ(z)rn|f′(rnϕ(z))||ϕ′(z)|.
Then we have
(40)S1=sup|ϕ(z)|>rNμ(z)|f′(ϕ(z))||ϕ′(z)|m×vlog(ϕ(z))|σϕ(z)′(ϕ(z))|vlog(ϕ(z))|σϕ(z)′(ϕ(z))|≤1Cσ∥f∥ℬlogsup|ϕ(z)|>rNμ(z)|σϕ(z)′(ϕ(z))||ϕ′(z)|≤1Cσsup|ϕ(z)|>rNsup|a|>rNμ(z)|σa′(ϕ(z))||ϕ′(z)|≤1Cσsup|a|>rN∥σa∘ϕ∥ℬμ,
where we have used the item (2) of Lemma 4 in the first inequality and the fact that ∥f∥ℬlog≤1 in the second one. Taking limit as N→∞, we obtain
(41)limsupn→∞S1≤1CσL.
In a similar way, we have
(42)S2≤1Cσ∥f∥ℬlogsup|ϕ(z)|>rNμ(z)|σϕ(z)′(ϕ(z))|m×|ϕ′(z)|rnvlog(ϕ(z))vlog(rnϕ(z))≤1Cσsup|a|>rN∥σa∘ϕ∥ℬμ,
where we have used Lemma 3 in the last inequality. Therefore
(43)∥Cϕ∥eℬlog→ℬμ≤2Cσlimsup|a|→1-∥σa∘ϕ∥ℬμ.
This completes the proof of the theorem.As an immediate consequence of Theorem2, we have the following corollary which extends the result due to Tjani in [12].Corollary 8.
The composition operatorCϕ is compact from ℬlog into ℬμ if and only if  ϕ∈ℬμ and
(44)lim|a|→1-∥σa∘ϕ∥μ=0.We want to finish this paper with the following question: Is the above corollary true for the composition operatorCϕ from ℬμ1 into ℬμ2, where μ1 and μ2 are weights defined on 𝔻?



---

*Source: 817278-2013-09-26.xml*