Stability of Admissible Functions

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 (𝑄_𝑝), and Hilbert Hardy class (𝐻²).

1. Introduction

We denote by 𝑈 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(𝑒^𝑖𝜃)|=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−₀^2𝜋𝑒^𝑖𝑡+𝑧𝑒^𝑖𝑡(𝑡)−𝑧𝑑𝜇,𝑧∈𝑈, (1.1) 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∶𝑋→𝑌 that satisfy𝑓(𝑥)≥1,when‖𝑥‖=1. (1.2)

If 𝑓 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 spaces 𝒢(𝑋,𝑌). The stability of analytic functions appears in Bloch class, little Bloch class, hyperbolic little Bloch class, extend Bloch class (𝑄_𝑝), 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 if𝑓_ℬ=𝑓(0)+sup_𝑧∈𝑈1−|𝑧|²𝑓^′(𝑧)<∞. (2.1) The space of all Bloch functions is denoted by ℬ. The little Bloch space ℬ₀ consists of those 𝑓∈ℬ such that lim_|𝑧|<11−|𝑧|²𝑓^′(𝑧)=0. (2.2) The hyperbolic Bloch class ℬ^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∶𝑈→𝑈 of the unit disk is given by |𝜑′|/(1−|𝜑|²). That is, 𝜑∈ℬ^ℎ if it is analytic andsup_𝑧∈𝑈1−|𝑧|²𝜑^′(𝑧)1−𝜑(𝑧)²<∞. (2.3) Similarly, we say 𝜑∈ℬ^ℎ₀, the hyperbolic little Bloch class, if 𝜑∈ℬ^ℎ andlim_|𝑧|=11−|𝑧|²𝜑^′(𝑧)1−𝜑(𝑧)²=0. (2.4) Note that, for the function 𝜑∶𝑈→𝑋, 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∶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. Let 𝑋=ℋ(𝑈) be a space of analytic functions in 𝑈 which satisfies ℬ⊂𝑋. Then, 𝒢(𝑈,𝑋) is stable.

Proof. Suppose that 𝑋=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. Let 𝑋=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 that 𝑋=ℋ(𝑈),𝑓∈𝒢(𝑋,𝑋), and 𝐼∶𝑈→𝑋 with 𝐼(0)=Θ. Then (see [3]) 𝑓(𝐼(𝑧))_𝑋<1⟹‖𝐼(𝑧)‖_𝑋<1, (2.5) 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 spaces ℬ₀ and ℬ^ℎ₀. An analytic self-map 𝜑 of 𝑈 induces a linear operator 𝐶_𝜑∶ℋ(𝑈)→ℋ(𝑈), defined by 𝐶_𝜑𝑓=𝑓∘𝜑. This operator is called the composition operator induced by 𝜑.

Recall that a linear operator 𝑇∶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 𝐶_𝜑𝑓, we have the following result.

Theorem 2.4. If 𝒢(ℬ₀,ℬ₀) is compact, then it is stable.

Proof. Assume the analytic self-map 𝜑 of 𝑈 and 𝑓∈ℬ₀; thus, we have the linear operator 𝐶_𝜑∶ℬ₀→ℬ₀, defined by 𝐶_𝜑𝑓=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. Let 𝜑 be holomorphic self-map of 𝑈 such that lim_|𝑧|=11−|𝑧|²𝜑^′(𝑧)1−𝜑(𝑧)²=0. (2.6) Then, 𝒢(ℬ₀,ℬ₀) is stable.

Proof. Assume the analytic self-map 𝜑 of 𝑈 and 𝑓∈ℬ₀; hence, in virtue of [5, Theorem  4.7], it is implied that the composition operator 𝐶_𝜑𝑓 on ℬ₀ is compact. Thus we pose that 𝒢(ℬ₀,ℬ₀) is stable.

Theorem 2.6. Consider 𝜑 is a holomorphic self-map of 𝑈, satisfying the following condition: for every 𝜖>0,thereexists0<𝑟<1 such that 1−|𝑧|²𝜑^′(𝑧)1−𝜑(𝑧)²<𝜖 (2.7) when |𝜑|>𝑟. Then, 𝒢(ℬ₀,ℬ₀) is stable.

Proof. Assume the analytic self-map 𝜑 of 𝑈 and 𝑓∈ℬ₀; hence, in virtue of [5, Theorem  4.8], it is yielded that the composition operator 𝐶_𝜑𝑓 on ℬ₀ is compact. Hence, we obtain that 𝒢(ℬ₀,ℬ₀) is stable.

Theorem 2.7. If 𝐶_𝜑𝑓 is weakly compact in ℬ₀, then 𝒢(ℬ₀,ℬ₀) is stable.

Proof. According to [5, Theorem  4.10], we have that 𝐶_𝜑𝑓 is compact in ℬ₀ and, consequently, 𝒢(ℬ₀,ℬ₀) is stable.

Theorem 2.8. Let 𝜑 be holomorphic self-map of 𝑈. If the function 𝜏_𝜑(𝑧)|𝑧|∶=1−²𝜑^′(𝑧)1−𝜑(𝑧)² (2.8) is bounded, then 𝒢(ℬ^ℎ₀,ℬ^ℎ₀) is stable.

Proof. Assume the analytic self-map 𝜑 of 𝑈 and 𝑓∈ℬ^ℎ₀. Since 𝜏_𝜑(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∈ℬ^ℎ₀, we have the desired result.

Theorem 2.9. If 𝜑∈ℬ^ℎ₀, then 𝒢(ℬ₀,ℬ₀) is stable.

Proof. Following [4], it will be shown that there are inner functions 𝜑∈ℬ^ℎ₀; then, 𝐶_𝜑∶ℬ₀→ℬ₀ is compact (see [4, 5]). Thus, 𝒢(ℬ₀,ℬ₀) is stable.

Theorem 2.10. Let 𝜑 be self-map in 𝑈 and 𝑤∶(0,1]→(0,∞) be continuous with lim_𝑡=0𝑤(𝑡)=0 satisfying lim_|𝑧|↑11−|𝑧|²𝜑^′(𝑧)𝑤1−𝜑(𝑧)²=0, (2.9) 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], 𝐶_𝜑∶ℬ₀→ℬ₀ is compact; hence, 𝒢(ℬ₀,ℬ₀) is stable.

Remark 2.11. The Schwarz-Pick Lemma implies
(i) 𝐶_𝜑mapsℬtoℬ;
(ii) 0≤|𝜏_𝜑(𝑧)|≤1;
(iii) 𝜑∈ℬ₀if𝐶_𝜑mapsℬ₀→ℬ₀andconversely,𝑓,𝜑∈ℬ₀⇒𝑓∘𝜑∈ℬ₀.

3. Stability of the Hilbert Hardy Space

In this section, we assume that 𝑓∈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 𝐶_𝜑𝑓=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 if𝐶_𝜑𝑓_𝑒=1+𝜑(0)𝜑(0,1−) (3.1) where |𝐶_𝜑𝑓|_𝑒 denotes the essential norm of 𝐶_𝜑𝑓.

Theorem 3.1. Let 𝜑 be self-map of 𝑈 and 𝑓∈𝐻². If (1.1) holds, then 𝒢(𝐻²,𝐻²) is stable.

Proof. Assume the analytic self-map 𝜑 of 𝑈 and 𝑓∈𝐻². Condition (1.1) implies that 𝜑 is an inner function. By setting 𝑔∶=𝐶_𝜑𝑓=𝑓∘𝜑 and, consequently, 𝑔∈𝐻², it is yielded that 𝒢(𝐻²,𝐻²) is stable.

Next we will show that the compactness of 𝐶_𝜑𝑓 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 𝐶_𝜑𝑓∶𝐻²→𝐻² is compact, then 𝒢(𝐻²,𝐻²) is stable.

Proof. Since 𝐶_𝜑∶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∶=𝐶_𝜑𝑓=𝑓∘𝜑, and, consequently, 𝑔∈𝐻², 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 𝐶_𝜑∶𝐻²→𝐻². Since 𝜑 has values never approach the boundary of 𝑈∶𝜑_∞=sup𝜑,𝑧∈𝑈<1, (3.2)𝐶_𝜑 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 𝐶_𝜑 is compact on 𝐻², then it is compact on 𝐻^𝑝 for all 0<𝑝<∞ (see [9, Theorem  6.1]).
(ii) 𝐶_𝜑 is compact on 𝐻^∞ if and only if ‖𝜑‖_∞<1 (see [10, Theorem  2.8]).

Theorem 3.5. If 𝜑∈ℬ^ℎ₀ is univalent then, 𝒢(𝐿^𝑝_𝑎,𝐻^𝑞),0<𝑝<𝑞<∞ is stable, where 𝐿^𝑝_𝑎,𝐻^𝑞 are the classical Bergman and Hardy spaces.

Proof. Since 𝜑 is univalent, 𝐶_𝜑∶𝐿^𝑝_𝑎→𝐻^𝑞 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(𝐿^𝑝_𝑎,𝐻^𝑞) is stable.

Next, we use the angular derivative criteria to discuss the stability of admissible functions. Recall that 𝜑 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. If 𝜑 satisfies both the angular derivative criteria and sup𝑛_𝜑(𝑤)∶𝑟<|𝑤|<1,0<𝑟<1<∞, (3.3) where 𝑛_𝜑(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 𝐶_𝜑 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 of 𝑄_𝑝 Class

For 0<𝑝<1, an analytic function 𝑓 in 𝑈 is said to belong to the space 𝑄_𝑝 ifsup_{𝑎∈𝑈𝑈}𝑓^′(𝑧)²𝑔(𝑧,𝑎)^𝑝𝑑𝐴(𝑧)<∞, (4.1) where 𝑑𝐴(𝑧)=𝑑𝑥𝑑𝑦=𝑟𝑑𝑟𝑑𝜃 is the Lebesgue area measure and 𝑔 denotes the Green function for the disk given by𝑔(𝑧;𝑎)=log1−𝑎𝑧𝑎−𝑧,𝑎,𝑧∈𝑈,𝑎≠𝑧. (4.2) The spaces 𝑄_𝑝 are conformally invariant. In [13], It was shown that 𝑄_𝑝=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,Λ^𝛼 is the Lipschitz space, consisting of those 𝑓∈ℋ(𝑈), which are continuous in 𝑈 and satisfy𝑓𝑧₁−𝑓𝑧₂≤𝐶𝑧₁−𝑧₂^𝛼,𝑧₁,𝑧₂∈𝑈, (4.3) for some 𝐶=𝐶(𝑓)>0. In this section, we will show the stability of functions belong to the spaces 𝑄₁ and Λ^𝛼.

Theorem 4.1. If 𝑓∈𝑄₁, 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∶=𝑓∘𝜑∈𝑄₁, even though 𝑓∈𝑄₁. Thus, 𝑄₁ is stable.

Theorem 4.2. If 𝑓∈Λ^𝛼,0<𝛼<1 such that 𝑓(𝑧)=𝑂(1−|𝑧|)^𝛼 (4.4) for some 𝑧, then 𝒢(Λ^𝛼,Λ^𝛼) 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∶=𝑓∘𝜑∈Λ^𝛼, even though 𝑓∈Λ^𝛼. Thus, Λ^𝛼 is stable.

5. Conclusion

From above, we conclude that the composition operator 𝐶_𝜑, 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 𝑄_𝑝-spaces and Lipschitz spaces.