# Stability of Admissible Functions

Authors: Rabha W. Ibrahim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

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).


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## Body

## 1. Introduction

We denote byU the unit disk {z∈ℂ:|z|<1} and by ℋ(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 form(1.1)S(z)=exp⁡(-∫02πeit+zeit-zdμ(t)),z∈U,
where μ is a finite positive Lebesgue measure.LetX and Y represent complex Banach spaces. The class of admissible functions 𝒢(X,Y), consists of those functions f:X→Y that satisfy(1.2)‖f(x)‖≥1,when‖x‖=1.Iff and g are analytic functions with f,g∈𝒢(X,Y), then f is said to be weakly subordinate to g, written as f≺wg if there exist analytic functions ϕ,ω:U→X, with ϕ an inner function (∥ϕ∥X≤1), so that f∘ϕ=g∘ω. A class 𝒞 of analytic functions in X is said to be stable if it is closed under weak subordination, that is, if f∈𝒞 whenever f and g are analytic functions in X with g∈𝒞 and f≺wg.By making use of the above concept of the weak subordination, we examine the stability for a class of admissible functions in complex Banach spaces𝒢(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

Iff is an analytic function in U, then f is said to be a Bloch function if(2.1)‖f‖B=|f(0)|+sup⁡z∈U⁡(1-|z|2)|f′(z)|<∞.
The space of all Bloch functions is denoted by ℬ. The little Bloch space ℬ0 consists of those f∈ℬ such that
(2.2)lim|z|<1(1-|z|2)|f′(z)|=0.
The hyperbolic Bloch class ℬh 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:U→U of the unit disk is given by |φ′|/(1-|φ|2). That is, φ∈ℬh if it is analytic and(2.3)sup⁡z∈U(1-|z|2)|φ′(z)|1-|φ(z)|2<∞.
Similarly, we say φ∈ℬ0h, the hyperbolic little Bloch class, if φ∈ℬh and(2.4)lim⁡|z|=1(1-|z|2)|φ′(z)|1-|φ(z)|2=0.
Note that, for the function φ:U→X, we replace |·| by ∥·∥X in the above definitions.Theorem 2.1.
LetX be a complex Banach space. If X contains all inner functions in U, then 𝒢(X,X) is stable.Proof.
Suppose thatX contains all inner functions I:U→X. Take g(z)=ϕ(z)=z and ω(z)=I(z)(z∈U). Then, ϕ and ω are inner functions and I=I∘ϕ=g∘ω. Hence, I≺wg,g∈X, and I∈X. Thus, X is stable.Theorem 2.2.
LetX=ℋ(U) be a space of analytic functions in U which satisfies ℬ⊂X. Then, 𝒢(U,X) is stable.Proof.
Suppose thatX=ℋ(U) and ℬ⊂X. Let f∈X,ℰ={m+ni:m,n∈ℤ} and F={z∈U:f(z)∈ℰ}. Since F is a countable subset of U, it has capacity zero and therefore the universal covering map I from U onto U∖F is an inner function (see, e.g., Chapter 2 of [1]). Set g=f∘I, then the image of g is contained in ℂ/ℰ. Consequently, see [2], g is a Bloch function. Since ℬ⊂X, we have that g=f∘I∈X even though f∈X. Thus, X is stable.Theorem 2.3.
LetX=ℋ(U) be a space of analytic functions in U and f∈𝒢(X,X). If I:U→X satisfying I(0)=Θ (the zero element in X) and ∥f(I(z))∥X<1, then 𝒢(X,X) is stable.Proof.
Assume thatX=ℋ(U),f∈𝒢(X,X), and I:U→X with I(0)=Θ. Then (see [3])
(2.5)‖f(I(z))‖X<1⟹‖I(z)‖X<1,
hence I is an inner function in X. By putting g:=f∘I, we obtain that g∈X even though f∈X. Thus, X is stable and consequently yields the stability of 𝒢(X,X).Next we discuss the stability of the spacesℬ0 and ℬ0h. An analytic self-map φ of U induces a linear operator Cφ:ℋ(U)→ℋ(U), defined by Cφf=f∘φ. This operator is called the composition operator induced by φ.Recall that a linear operatorT:X→Y 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.
If𝒢(ℬ0,ℬ0) is compact, then it is stable.Proof.
Assume the analytic self-mapφ of U and f∈ℬ0; thus, we have the linear operator Cφ:ℬ0→ℬ0, defined by Cφf=f∘φ:=g. By the assumption, we obtain that g is compact function in ℬ0. Hence, φ is an inner function [4, Corollary  1.3] which implies the stability of 𝒢(ℬ0,ℬ0).Theorem 2.5.
Letφ be holomorphic self-map of U such that
(2.6)lim⁡|z|=1(1-|z|2)|φ′(z)|1-|φ(z)|2=0.
Then, 𝒢(ℬ0,ℬ0) is stable.Proof.
Assume the analytic self-mapφ of U and f∈ℬ0; hence, in virtue of [5, Theorem  4.7], it is implied that the composition operator Cφf on ℬ0 is compact. Thus we pose that 𝒢(ℬ0,ℬ0) is stable.Theorem 2.6.
Considerφ is a holomorphic self-map of U, satisfying the following condition: for every ϵ>0,thereexists0<r<1 such that
(2.7)(1-|z|2)|φ′(z)|1-|φ(z)|2<ϵ
when |φ|>r. Then, 𝒢(ℬ0,ℬ0) is stable.Proof.
Assume the analytic self-mapφ of U and f∈ℬ0; hence, in virtue of [5, Theorem  4.8], it is yielded that the composition operator Cφf on ℬ0 is compact. Hence, we obtain that 𝒢(ℬ0,ℬ0) is stable.Theorem 2.7.
IfCφf is weakly compact in ℬ0, then 𝒢(ℬ0,ℬ0) is stable.Proof.
According to [5, Theorem  4.10], we have that Cφf is compact in ℬ0 and, consequently, 𝒢(ℬ0,ℬ0) is stable.Theorem 2.8.
Letφ be holomorphic self-map of U. If the function
(2.8)τφ(z):=(1-|z|2)|φ′(z)|1-|φ(z)|2
is bounded, then 𝒢(ℬ0h,ℬ0h) is stable.Proof.
Assume the analytic self-mapφ of U and f∈ℬ0h. 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:=f∘φ, where g∈ℬ0h, we have the desired result.Theorem 2.9.
Ifφ∈ℬ0h, then 𝒢(ℬ0,ℬ0) is stable.Proof.
Following [4], it will be shown that there are inner functions φ∈ℬ0h; then, Cφ:ℬ0→ℬ0 is compact (see [4, 5]). Thus, 𝒢(ℬ0,ℬ0) is stable.Theorem 2.10.
Letφ be self-map in U and w:(0,1]→(0,∞) be continuous with limt=0w(t)=0 satisfying
(2.9)lim⁡|z|↑1(1-|z|2)|φ′(z)|w(1-|φ(z)|2)=0,
then 𝒢(ℬ0,ℬ0) is stable.Proof.
According to [5, Theorem  5.15], we pose that φ is inner. Thus, in view of [4, Corollary  1.3], Cφ:ℬ0→ℬ0 is compact; hence, 𝒢(ℬ0,ℬ0) is stable.Remark 2.11.
The Schwarz-Pick Lemma implies
(i)Cφmapsℬtoℬ;
(ii)0≤|τφ(z)|≤1;
(iii)φ∈ℬ0ifCφmapsℬ0→ℬ0andconversely,f,φ∈ℬ0⇒f∘φ∈ℬ0.


## 3. Stability of the Hilbert Hardy Space

In this section, we assume thatf∈H2, 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=f∘φ on H2. Moreover, Joel Shapiro obtained the following characterization of inner functions [6]: the function φ:U→U is inner if and only if(3.1)‖Cφf‖e=1+|φ(0)|1-|φ(0)|,
where |Cφf|e denotes the essential norm of Cφf.Theorem 3.1.
Letφ be self-map of U and f∈H2. If (1.1) holds, then 𝒢(H2,H2) is stable.Proof.
Assume the analytic self-mapφ of U and f∈H2. Condition (1.1) implies that φ is an inner function. By setting g:=Cφf=f∘φ and, consequently, g∈H2, it is yielded that 𝒢(H2,H2) is stable.Next we will show that the compactness ofCφf 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 operatorCφf:H2→H2 is compact, then 𝒢(H2,H2) is stable.Proof.
SinceCφ:H2→H2 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=f∘φ, and, consequently, g∈H2, it is yielded that 𝒢(H2,H2) is stable.Theorem 3.3.
Ifφ has values never approach the boundary of U, then 𝒢(H2,H2) is stable.Proof.
Assume the composition operatorCφ:H2→H2. Since φ has values never approach the boundary of U:(3.2)‖φ‖∞=sup⁡{|φ|,z∈U}<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 ifCφ 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.
Ifφ∈ℬ0h is univalent then, 𝒢(Lap,Hq),0<p<q<∞ is stable, where Lap,Hq are the classical Bergman and Hardy spaces.Proof.
Sinceφ is univalent, Cφ:Lap→Hq 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 thatφ has angular derivative at ζ∈∂U if the nontangential limw=f(ζ)∈∂U exists and if (f(z)-f(ζ))/(z-ζ) converges to some μ∈ℂ as z→ζ nontangentially.Theorem 3.6.
Ifφ satisfies both the angular derivative criteria and
(3.3)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.


## 4. Stability ofQp Class

For0<p<1, an analytic function f in U is said to belong to the space Qp if(4.1)sup⁡a∈U∫U|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 by(4.2)g(z;a)=log⁡1-a¯za-z,a,z∈U,a≠z.
The spaces Qp are conformally invariant. In [13], It was shown that Qp=ℬ for all p, while Q1=BMOA, the space of those f∈H1 whose boundary values have bounded mean oscillation on ∂U (see [14]). For 0<α<1,Λα is the Lipschitz space, consisting of those f∈ℋ(U), which are continuous in U and satisfy(4.3)|f(z1)-f(z2)|≤C|z1-z2|α,z1,z2∈U,
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.
Iff∈Q1, then 𝒢(Q1,Q1) is stable.Proof.
In the similar manner of Theorem2.2, we pose an inner function φ on U. Now, in view of [15, Theorem  H], yields g:=f∘φ∈Q1, even though f∈Q1. Thus, Q1 is stable.Theorem 4.2.
Iff∈Λα,0<α<1 such that
(4.4)|f(z)|=O((1-|z|)α)
for some z, then 𝒢(Λα,Λα) is stable.Proof.
Again as in Theorem2.2, we obtain an inner function φ on U. Now in view of [15, Theorem  K], yields g:=f∘φ∈Λα, even though f∈Λα. Thus, Λα is stable.


## 5. Conclusion

From above, we conclude that the composition operatorCφ, 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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