IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation34289510.1155/2011/342895342895Research ArticleStability of Admissible FunctionsIbrahimRabha W.ChoNakSchool of Mathematical SciencesFaculty of Sciences and TechnologyUKM, 43600 BangiMalaysiaukm.my20111592011201102062011160720112011Copyright © 2011 Rabha W. Ibrahim.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 (Qp), and Hilbert Hardy class (H2).

1. Introduction

We denote by U the unit disk {z:|z|<1} and by (U) the space of all analytic functions in U. A function I, analytic in U, is said to be an inner function if and only if |I(z)|1 such that |I(eiθ)|=1 almost everywhere. We recall that an inner function I can be factored in the form I=BS where B is a Blaschke product and S is a singular inner function takes the formS(z)=exp(-02πeit+zeit-zdμ(t)),zU, where μ is a finite positive Lebesgue measure.

Let X and Y represent complex Banach spaces. The class of admissible functions 𝒢(X,Y), consists of those functions f:XY that satisfyf(x)1,when  x=1.

If f and g are analytic functions with f,g𝒢(X,Y), then f is said to be weakly subordinate to g, written as fwg if there exist analytic functions ϕ,ω:UX, with ϕ an inner function (ϕX1), so that fϕ=gω. A class 𝒞 of analytic functions in X is said to be stable if it is closed under weak subordination, that is, if f𝒞 whenever f and g are analytic functions in X with g𝒞 and fwg.

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 𝒢(X,Y). The stability of analytic functions appears in Bloch class, little Bloch class, hyperbolic little Bloch class, extend Bloch class (Qp), and Hilbert Hardy class (H2).

2. Stability of Bloch Classes

If f is an analytic function in U, then f is said to be a Bloch function iffB=|f(0)|+supzU(1-|z|2)|f(z)|<. The space of all Bloch functions is denoted by . The little Bloch space 0 consists of those f such that lim|z|<1(1-|z|2)|f(z)|=0. The hyperbolic Bloch class h 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 φ:UU of the unit disk is given by |φ|/(1-|φ|2). That is, φh if it is analytic andsupzU(1-|z|2)|φ(z)|1-|φ(z)|2<. Similarly, we say φ0h, the hyperbolic little Bloch class, if φh andlim|z|=1(1-|z|2)|φ(z)|1-|φ(z)|2=0. Note that, for the function φ:UX, we replace |·| by ·X in the above definitions.

Theorem 2.1.

Let X be a complex Banach space. If X contains all inner functions in U, then 𝒢(X,X) is stable.

Proof.

Suppose that X contains all inner functions I:UX. Take g(z)=ϕ(z)=z and ω(z)=I(z)  (zU). Then, ϕ and ω are inner functions and I=Iϕ=gω. Hence, Iwg,  gX, and IX. Thus, X is stable.

Theorem 2.2.

Let X=(U) be a space of analytic functions in U which satisfies X. Then, 𝒢(U,X) is stable.

Proof.

Suppose that X=(U) and X. Let fX,  ={m+ni:m,n} and F={zU:f(z)}. Since F is a countable subset of U, it has capacity zero and therefore the universal covering map I from U onto UF is an inner function (see, e.g., Chapter 2 of ). Set g=fI, then the image of g is contained in /. Consequently, see , g is a Bloch function. Since X, we have that g=fIX even though fX. Thus, X is stable.

Theorem 2.3.

Let X=(U) be a space of analytic functions in U and f𝒢(X,X). If I:UX satisfying I(0)=Θ (the zero element in X) and f(I(z))X<1, then 𝒢(X,X) is stable.

Proof.

Assume that X=(U),f𝒢(X,X), and I:UX with I(0)=Θ. Then (see ) f(I(z))X<1I(z)X<1, hence I is an inner function in X. By putting g:=fI, we obtain that gX even though fX. Thus, X is stable and consequently yields the stability of 𝒢(X,X).

Next we discuss the stability of the spaces 0 and 0h. An analytic self-map φ of U induces a linear operator Cφ:(U)(U), defined by Cφf=fφ. This operator is called the composition operator induced by φ.

Recall that a linear operator T:XY is said to be bounded if the image of a bounded set in X is a bounded subset of Y, while T is compact if it takes bounded sets to sets with compact closure. Furthermore, if T is a bounded linear operator, then it is called weakly compact, if T takes bounded sets in X to relatively weakly compact sets in Y. By using the operator Cφf, we have the following result.

Theorem 2.4.

If 𝒢(0,0) is compact, then it is stable.

Proof.

Assume the analytic self-map φ of U and f0; thus, we have the linear operator Cφ:00, defined by Cφf=fφ:=g. By the assumption, we obtain that g is compact function in 0. Hence, φ is an inner function [4, Corollary  1.3] which implies the stability of 𝒢(0,0).

Theorem 2.5.

Let φ be holomorphic self-map of U such that lim|z|=1(1-|z|2)|φ(z)|1-|φ(z)|2=0. Then, 𝒢(0,0) is stable.

Proof.

Assume the analytic self-map φ of U and f0; hence, in virtue of [5, Theorem  4.7], it is implied that the composition operator Cφf on 0 is compact. Thus we pose that 𝒢(0,0) is stable.

Theorem 2.6.

Consider φ is a holomorphic self-map of U, satisfying the following condition: for every ϵ>0,  there  exists  0<r<1 such that (1-|z|2)|φ(z)|1-|φ(z)|2<ϵ when |φ|>r. Then, 𝒢(0,0) is stable.

Proof.

Assume the analytic self-map φ of U and f0; hence, in virtue of [5, Theorem  4.8], it is yielded that the composition operator Cφf on 0 is compact. Hence, we obtain that 𝒢(0,0) is stable.

Theorem 2.7.

If Cφf is weakly compact in 0, then 𝒢(0,0) is stable.

Proof.

According to [5, Theorem  4.10], we have that Cφf is compact in 0 and, consequently, 𝒢(0,0) is stable.

Theorem 2.8.

Let φ be holomorphic self-map of U. If the function τφ(z):=(1-|z|2)|φ(z)|1-|φ(z)|2 is bounded, then 𝒢(0h,0h) is stable.

Proof.

Assume the analytic self-map φ of U and f0h. Since τφ(z) is bounded, then in virtue of [4, Theorem  1.2], it is yielded that φ is an inner function. By putting g:=fφ, where g0h, we have the desired result.

Theorem 2.9.

If φ0h, then 𝒢(0,0) is stable.

Proof.

Following , it will be shown that there are inner functions φ0h; then, Cφ:00 is compact (see [4, 5]). Thus, 𝒢(0,0) is stable.

Theorem 2.10.

Let φ be self-map in U and w:(0,1](0,) be continuous with limt=0w(t)=0 satisfying lim|z|1(1-|z|2)|φ(z)|w(1-|φ(z)|2)=0, then 𝒢(0,0) is stable.

Proof.

According to [5, Theorem  5.15], we pose that φ is inner. Thus, in view of [4, Corollary  1.3], Cφ:00 is compact; hence, 𝒢(0,0) is stable.

Remark 2.11.

The Schwarz-Pick Lemma implies

(i) Cφ  maps    to  ;

(ii)   0|τφ(z)|1;

(iii)   φ0  if  Cφ  maps  00  and  conversely,f,φ0fφ0.

3. Stability of the Hilbert Hardy Space

In this section, we assume that fH2, where H2 is the Hilbert Hardy space on U, that is, the set of all analytic functions on U with square summable Taylor coefficients. It is well known that each such φ (self-map in U) induces a bounded composition operator Cφf=fφ on H2. Moreover, Joel Shapiro obtained the following characterization of inner functions : the function φ:UU is inner if and only ifCφfe=1+|φ(0)|1-|φ(0)|, where |Cφf|e denotes the essential norm of Cφf.

Theorem 3.1.

Let φ be self-map of U and fH2. If (1.1) holds, then 𝒢(H2,H2) is stable.

Proof.

Assume the analytic self-map φ of U and fH2. Condition (1.1) implies that φ is an inner function. By setting g:=Cφf=fφ and, consequently, gH2, it is yielded that 𝒢(H2,H2) is stable.

Next we will show that the compactness of Cφf introduces the stability of 𝒢(H2,H2). Two positive (or complex) measures μ and ν defined on a measurable space (Ω,Σ) are called singular if there exist two disjoint sets A and B in Σ whose union is Ω such that μ is zero on all measurable subsets of B while ν is zero on all measurable subsets of A.

Theorem 3.2.

If the composition operator Cφf:H2H2 is compact, then 𝒢(H2,H2) is stable.

Proof.

Since Cφ:H2H2 is compact, all the Aleksandrov measures of φ are singular absolutely continuous with respect to the arc-length measure (see [7, 8]). Thus, in view of [4, Remark  1], φ is inner. By letting g:=Cφf=fφ, and, consequently, gH2, it is yielded that 𝒢(H2,H2) is stable.

Theorem 3.3.

If φ has values never approach the boundary of U, then 𝒢(H2,H2) is stable.

Proof.

Assume the composition operator Cφ:H2H2. Since φ has values never approach the boundary of U:φ=sup{|φ|,  zU}<1,Cφ is compact on H2 (see [9, 10]). Hence, φ is an inner function and 𝒢(H2,H2) is stable.

Remark 3.4.

(i) It is well known that if Cφ is compact on H2, then it is compact on Hp for all 0<p< (see [9, Theorem  6.1]).

(ii) Cφ is compact on H if and only if φ<1 (see [10, Theorem  2.8]).

Theorem 3.5.

If φ0h is univalent then, 𝒢(Lap,Hq),  0<p<q< is stable, where Lap,Hq are the classical Bergman and Hardy spaces.

Proof.

Since φ is univalent, Cφ:LapHq is compact for all 0<p<q<(see [11, Theorem  6.4]). In view of Remark 3.4, we obtain that φ is an inner function; hence, 𝒢(Lap,Hq) is stable.

Next, we use the angular derivative criteria to discuss the stability of admissible functions. Recall that φ has angular derivative at ζU if the nontangential limw=f(ζ)U exists and if (f(z)-f(ζ))/(z-ζ) converges to some μ as zζ nontangentially.

Theorem 3.6.

If φ satisfies both the angular derivative criteria and sup{nφ(w):r<|w|<1,0<r<1}<, where nφ(w) is the number of points in φ-1(w) with multiplicity counted, then 𝒢(H2,H2) is stable.

Proof.

According to [12, Corollary  3.6], we have that Cφ is compact on H2. Again in view of Remark 3.4, we obtain that φ is inner and, consequently, 𝒢(H2,H2) is stable.

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For 0<p<1, an analytic function f in U is said to belong to the space Qp ifsupaUU|f(z)|2g(z,a)pdA(z)<, where dA(z)=dxdy=rdrdθ is the Lebesgue area measure and g denotes the Green function for the disk given byg(z;a)=log1-a¯za-z,        a,zU,  az. The spaces Qp are conformally invariant. In , It was shown that Qp= for all p, while Q1=BMOA, the space of those fH1 whose boundary values have bounded mean oscillation on U (see ). For 0<α<1,  Λα is the Lipschitz space, consisting of those f(U), which are continuous in U and satisfy|f(z1)-f(z2)|C|z1-z2|α,        z1,z2U, for some C=C(f)>0. In this section, we will show the stability of functions belong to the spaces Q1 and Λα.

Theorem 4.1.

If fQ1, then 𝒢(Q1,Q1) is stable.

Proof.

In the similar manner of Theorem 2.2, we pose an inner function φ on U. Now, in view of [15, Theorem  H], yields g:=fφQ1, even though fQ1. Thus, Q1 is stable.

Theorem 4.2.

If fΛα,  0<α<1 such that |f(z)|=O((1-|z|)α) for some z, then 𝒢(Λα,Λα) is stable.

Proof.

Again as in Theorem 2.2, we obtain an inner function φ on U. Now in view of [15, Theorem  K], yields g:=fφΛα, even though fΛα. Thus, Λα is stable.

5. Conclusion

From above, we conclude that the composition operator Cφ, 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 φ is an inner function on the unit disk U. Furthermore, weakly compactness imposed the stability of Bloch spaces. In addition, noncompactness leaded to the stability for some spaces such as Qp-spaces and Lipschitz spaces.

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