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Stability of Admissible Functions
By using the concept of the weak subordination, we examine the stability (a class of analytic functions in the unit disk is said to be stable if it is closed under weak subordination) for a class of admissible functions in complex Banach spaces. The stability of analytic functions in the following classes is discussed: Bloch class, little Bloch class, hyperbolic little Bloch class, extend Bloch class , and Hilbert Hardy class .
We denote by the unit disk and by the space of all analytic functions in . A function , analytic in , is said to be an inner function if and only if such that almost everywhere. We recall that an inner function can be factored in the form where is a Blaschke product and is a singular inner function takes the form where is a finite positive Lebesgue measure.
Let and represent complex Banach spaces. The class of admissible functions , consists of those functions that satisfy
If and are analytic functions with , then is said to be weakly subordinate to , written as if there exist analytic functions , with an inner function , so that . A class of analytic functions in is said to be stable if it is closed under weak subordination, that is, if whenever and are analytic functions in with and .
By making use of the above concept of the weak subordination, we examine the stability for a class of admissible functions in complex Banach spaces . The stability of analytic functions appears in Bloch class, little Bloch class, hyperbolic little Bloch class, extend Bloch class , and Hilbert Hardy class .
2. Stability of Bloch Classes
If f is an analytic function in , then is said to be a Bloch function if The space of all Bloch functions is denoted by . The little Bloch space consists of those such that The hyperbolic Bloch class is defined by using the hyperbolic derivative in place of the ordinary derivative in the definition of the Bloch space, where the hyperbolic derivative of an analytic self-map of the unit disk is given by . That is, if it is analytic and Similarly, we say , the hyperbolic little Bloch class, if and Note that, for the function , we replace by in the above definitions.
Theorem 2.1. Let be a complex Banach space. If contains all inner functions in , then is stable.
Proof. Suppose that contains all inner functions . Take and . Then, and are inner functions and . Hence, , and . Thus, X is stable.
Theorem 2.2. Let be a space of analytic functions in which satisfies . Then, is stable.
Proof. Suppose that and . Let and . Since is a countable subset of , it has capacity zero and therefore the universal covering map from onto is an inner function (see, e.g., Chapter 2 of ). Set , then the image of is contained in . Consequently, see , is a Bloch function. Since , we have that even though . Thus, is stable.
Theorem 2.3. Let be a space of analytic functions in and . If satisfying (the zero element in ) and , then is stable.
Proof. Assume that , and with . Then (see ) hence is an inner function in . By putting , we obtain that even though . Thus, is stable and consequently yields the stability of .
Next we discuss the stability of the spaces and . An analytic self-map of induces a linear operator , defined by . This operator is called the composition operator induced by .
Recall that a linear operator is said to be bounded if the image of a bounded set in is a bounded subset of , while is compact if it takes bounded sets to sets with compact closure. Furthermore, if is a bounded linear operator, then it is called weakly compact, if takes bounded sets in to relatively weakly compact sets in . By using the operator , we have the following result.
Theorem 2.4. If is compact, then it is stable.
Proof. Assume the analytic self-map of and ; thus, we have the linear operator , defined by . By the assumption, we obtain that is compact function in . Hence, is an inner function [4, Corollary 1.3] which implies the stability of .
Theorem 2.5. Let be holomorphic self-map of such that Then, is stable.
Proof. Assume the analytic self-map of and ; hence, in virtue of [5, Theorem 4.7], it is implied that the composition operator on is compact. Thus we pose that is stable.
Theorem 2.6. Consider is a holomorphic self-map of , satisfying the following condition: for every such that when . Then, is stable.
Proof. Assume the analytic self-map of and ; hence, in virtue of [5, Theorem 4.8], it is yielded that the composition operator on is compact. Hence, we obtain that is stable.
Theorem 2.7. If is weakly compact in , then is stable.
Proof. According to [5, Theorem 4.10], we have that is compact in and, consequently, is stable.
Theorem 2.8. Let be holomorphic self-map of . If the function is bounded, then is stable.
Proof. Assume the analytic self-map of and . Since is bounded, then in virtue of [4, Theorem 1.2], it is yielded that is an inner function. By putting , where , we have the desired result.
Theorem 2.9. If , then is stable.
Proof. Following , it will be shown that there are inner functions ; then, is compact (see [4, 5]). Thus, is stable.
Theorem 2.10. Let be self-map in and be continuous with satisfying then is stable.
Proof. According to [5, Theorem 5.15], we pose that is inner. Thus, in view of [4, Corollary 1.3], is compact; hence, is stable.
Remark 2.11. The Schwarz-Pick Lemma implies
3. Stability of the Hilbert Hardy Space
In this section, we assume that , where is the Hilbert Hardy space on , that is, the set of all analytic functions on with square summable Taylor coefficients. It is well known that each such (self-map in ) induces a bounded composition operator on . Moreover, Joel Shapiro obtained the following characterization of inner functions : the function is inner if and only if where denotes the essential norm of .
Theorem 3.1. Let be self-map of and . If (1.1) holds, then is stable.
Proof. Assume the analytic self-map of and . Condition (1.1) implies that is an inner function. By setting and, consequently, , it is yielded that is stable.
Next we will show that the compactness of introduces the stability of . Two positive (or complex) measures and defined on a measurable space are called singular if there exist two disjoint sets and in whose union is such that is zero on all measurable subsets of while is zero on all measurable subsets of .
Theorem 3.2. If the composition operator is compact, then is stable.
Proof. Since is compact, all the Aleksandrov measures of are singular absolutely continuous with respect to the arc-length measure (see [7, 8]). Thus, in view of [4, Remark 1], is inner. By letting , and, consequently, , it is yielded that is stable.
Theorem 3.3. If has values never approach the boundary of , then is stable.
Proof. Assume the composition operator . Since has values never approach the boundary of is compact on (see [9, 10]). Hence, is an inner function and is stable.
Remark 3.4. (i) It is well known that if is compact on , then it is compact on for all (see [9, Theorem 6.1]).
(ii) is compact on if and only if (see [10, Theorem 2.8]).
Theorem 3.5. If is univalent then, is stable, where are the classical Bergman and Hardy spaces.
Proof. Since is univalent, is compact for all (see [11, Theorem 6.4]). In view of Remark 3.4, we obtain that is an inner function; hence, is stable.
Next, we use the angular derivative criteria to discuss the stability of admissible functions. Recall that has angular derivative at if the nontangential exists and if converges to some as nontangentially.
Theorem 3.6. If satisfies both the angular derivative criteria and where is the number of points in with multiplicity counted, then is stable.
Proof. According to [12, Corollary 3.6], we have that is compact on . Again in view of Remark 3.4, we obtain that is inner and, consequently, is stable.
4. Stability of Class
For , an analytic function in is said to belong to the space if where is the Lebesgue area measure and denotes the Green function for the disk given by The spaces are conformally invariant. In , It was shown that for all , while , the space of those whose boundary values have bounded mean oscillation on (see ). For is the Lipschitz space, consisting of those , which are continuous in and satisfy for some . In this section, we will show the stability of functions belong to the spaces and .
Theorem 4.1. If , then is stable.
Proof. In the similar manner of Theorem 2.2, we pose an inner function on . Now, in view of [15, Theorem H], yields , even though . Thus, is stable.
Theorem 4.2. If such that for some , then is stable.
Proof. Again as in Theorem 2.2, we obtain an inner function on . Now in view of [15, Theorem K], yields , even though . Thus, is stable.
From above, we conclude that the composition operator , of admissible functions in different complex Banach spaces, plays an important role in stability of these spaces. It was shown that the compactness of this operator implied the stability, when is an inner function on the unit disk . Furthermore, weakly compactness imposed the stability of Bloch spaces. In addition, noncompactness leaded to the stability for some spaces such as -spaces and Lipschitz spaces.
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