Natural metrics and composition operators in generalized hyperbolic function spaces
© El-Sayed Ahmed; licensee Springer 2012
Received: 29 March 2012
Accepted: 31 May 2012
Published: 31 August 2012
In this paper, we define some generalized hyperbolic function classes. We also introduce natural metrics in the generalized hyperbolic -Bloch and in the generalized hyperbolic classes. These classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, boundedness and compactness the composition operators acting from the generalized hyperbolic -Bloch class to the class are characterized by conditions depending on an analytic self-map .
respectively. Now, we will give the following definition:
Remark 1.1 The definition of -Bloch spaces is introduced in the present paper for the first time. One should note that, if we put in Definition 1.1, we will obtain the spaces and .
Remark 1.2-Bloch space is very useful in some calculations in this paper and it can be also used to study some other operators like integral operators (see ).
If is a metric space, we denote the open and closed balls with center x and radius by and , respectively. The well-known hyperbolic derivative is defined by of and the hyperbolic distance is given by between and zero.
The Schwarz-Pick lemma implies for all with , and therefore, the hyperbolic α-Bloch classes are of interest only when .
It is obvious that is not a linear space since the sum of two functions in does not necessarily belong to .
Now, let , we define the hyperbolic derivative by of . When , we obtain the usual hyperbolic derivative as defined above.
Remark 1.3 The Schwarz-Pick lemma implies that for all with and therefore, the generalized hyperbolic -classes are of interest only when . Also for all , and hence, the generalized hyperbolic -classes will be considered when .
For any holomorphic self-mapping ϕ of , the symbol ϕ induces a linear composition operator from or into itself. The study of a composition operator acting on the spaces of analytic functions has engaged many analysts for many years (see, e.g., [1–8, 11, 13, 16] and others).
Yamashita was probably the first to consider systematically hyperbolic function classes. He introduced and studied hyperbolic Hardy, BMOA and Dirichlet classes in [18–20] and others. More recently, Smith studied inner functions in the hyperbolic little Bloch-class , and the hyperbolic counterparts of the spaces were studied by Li in  and Li et al. in . Further, hyperbolic classes and composition operators were studied by Pérez-González et al. in .
In this paper, we will study the generalized hyperbolic -Bloch classes and the hyperbolic type classes. We will also give some results to characterize Lipschitz continuous and compact composition operators mapping from the generalized hyperbolic -Bloch class to classes by conditions depending on the symbol ϕ only. Thus, the results are generalizations of the recent results of Pérez-González, Rättyä and Taskinen .
Recall that a linear operator is said to be bounded if there exists a constant such that for all maps . By elementary functional analysis, a linear operator between normed spaces is bounded if and only if it is continuous, and the boundedness is trivially also equivalent to the Lipschitz-continuity. Moreover, is said to be compact if it takes bounded sets in X to sets in Y which have compact closure. For Banach spaces X and Y contained in or , is compact if and only if for each bounded sequence , the sequence contains a subsequence converging to a function .
Throughout this paper, C stands for absolute constants which may indicate different constants from one occurrence to the next.
The following lemma follows by standard arguments similar to those outlined in . Hence we omit the proof.
Lemma 1.1 Assume ϕ is a holomorphic mapping frominto itself. Let, and. Thenis compact if and only if for any bounded sequencewhich converges to zero uniformly on compact subsets ofas, we have.
Using the standard arguments similar to those outlined in Lemma 1 of , we have the following lemma:
2 Natural metrics in and classes
In this section we introduce natural metrics on generalized hyperbolic α-Bloch classes and the classes .
Now, we give a characterization of the complete metric space .
Proposition 2.1 The classequipped with the metricis a complete metric space. Moreover, is a closed (and therefore complete) subspace of.
Moreover, for all .
Thus, as desired. Moreover, (2) and the completeness of the usual -Bloch imply that converges to f with respect to the metric d. The second part of the assertion follows by (2). □
Next, we give a characterization of the complete metric space .
Proposition 2.2 The classequipped with the metricis a complete metric space. Moreover, is a closed (and therefore complete) subspace of.
Proof For , then clearly
Hence, d is metric on .
and thus . We also find that with respect to the metric of . The second part of the assertion follows by (3). □
3 Lipschitz continuous and compactness of
is Lipschitz continuous;
This estimate together with the Fatou’s lemma implies (iii).
Hence, it follows that (i) holds.
for which the assertion (iii) follows.
Thus is Lipschitz continuous and the proof is completed. □
By elementary functional analysis, a linear operator between normed spaces is bounded if and only if it is continuous, and the boundedness is trivially also equivalent to the Lipschitz-continuity. So, our result for composition operators in hyperbolic spaces is the correct and natural generalization of the linear operator theory.
Recall that a composition operator is compact if it maps any ball in onto a precompact set in .
The following observation is sometimes useful.
Proposition 3.1 Let, and. Assume that ϕ is a holomorphic mapping frominto itself. Ifis compact, it maps closed balls onto compact sets.
Proof If is a closed ball and belongs to the closure of , we can find a sequence such that converges to as . But is a normal family, hence it has a subsequence converging uniformly on the compact subsets of to an analytic function f. As in earlier arguments of Proposition 2.1 in , we get a positive estimate which shows that f must belong to the closed ball B. On the other hand, also the sequence converges uniformly on compact subsets to an analytic function, which is . We get , i.e., g belongs to . Thus, this set is closed and also compact. □
Compactness of composition operators can be characterized in full analogy with the linear case.
- (i)is compact.(ii)
Proof We first assume that (ii) holds. Let , where and , be a closed ball, and let be any sequence. We show that its image has a convergent subsequence in , which proves the compactness of by definition.
where C is the bound obtained from (iii) of Theorem 3.1. Combining (6), (7), (8) and (9), we deduce that in .
Then the sequence belongs to the ball .
From (11) and (13), the condition (ii) follows. This completes the proof. □
The proof of Proposition 2.2 implies the following corollary:
Moreover, the proofs of Theorems 3.1 and 3.2 yield the following result:
is Lipschitz continuous;
- (ii)is compact;(iii)
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