Certain Classes of Operators on Some Weighted Hyperbolic Function Spaces

In this paper, some classes of concerned multiplication operators consisting of analytic and hyperbolic functions are deﬁned and considered. Furthermore, some properties such as boundedness and compactness of the new operators are discussed. Finally, a general class of weighted hyperbolic Bloch functions is characterized by metric spaces.


Introduction
Complex function spaces are one of the interesting core subjects in mathematical analysis. is subject has a lot of various generalizations with many joyful branches. e study of hyperbolic function spaces has been at the rigorous research activity. e presented paper deals with discussion of some hyperbolic function classes. e known open-unit disc that defined in the concerned complex plane C and its specific boundary are symbolized by Δ � z: |z| < 1 { } and zΔ, respectively. e specific space of all holomorphic functions in Δ is denoted by H(Δ). Also, assuming that B(Δ) defines the subset of H(Δ), which contains those functions f ∈ H(Δ) with |f(z)| < 1 for all concerned points z ∈ Δ.
A concerned function f ∈ B(Δ) is said to belong to the specific hyperbolic α-Bloch class B * ω,α , when e little specific hyperbolic Bloch-type class B * ω,α,0 One of the major aims of this study is to give emerging treatments of some properties of a certain class of operators acting between hyperbolic Bloch functions using the concerned framework of hyperbolic spaces, which is based on the concerned technique of their specific functions. e multiplication operator between two different types of functions, one is analytic and the other is hyperbolic, is treated in this paper. Dealing with corresponding concerned spaces of weighted functions on the disk Δ, essential properties of the new type of operators are discussed. ere are some certain attempts to study hyperbolic function spaces (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and others). Most research studies were on composition operators. In this paper, by defining some new classes of operators, we will study hyperbolic logarithmic Bloch functions.

Boundedness and Compactness
Assuming f(z) and g(z) are two functions in H(Δ), the following operators are Further, the operator is one of the concerned aims in this study.

Remark 1.
e operators F(f(z)) and L g f(z) are introduced for the first time to study hyperbolic function classes in this paper.
An interesting approach of unified criteria for boundedness and compactness properties of new type operators acting between hyperbolic logarithmic Bloch spaces is investigated in the present section.
By using concerned Lemma 1 and considering the operator L g for g ∈ B * ω,α, ln ,0 using Lemma 2, we infer that g ∈ B * ω,α,0 . Hence, erefore, by (19), for every f ∈ B ω , we have that Journal of Mathematics (31) us, L g : B * ω ⟶ B * ω,α,0 is actually bounded. To prove that the concerned operator L g is actually compact, assume that f n ⊂ B * ω is defined such that ‖f n ‖ B * ω ≤ 1. We have to clear that L g f n has a concerned subsequence which is converging on B * ω,α,0 . Using (19), we can find a concerned subsequence of the sequence f n which converges uniformly on concerned compact subsets on the disc Δ to the concerned function f. For this purpose, we suppose that the concerned sequence f n converges also to the concerned function f. For any fixed z ∈ Δ, we infer that * ω,α,0 , and it is enough to be clear that Because g ∈ B * ω,,α, ln , we deduce for every ε > 0 and we can find an R ∈ (0, 1), for which

(34)
Because the sequence f n converges to the function f uniformly on each compact subset on Δ, we deduce that there exists an N > 0, for every n > N and every z ∈ Δ, |f n (z) − f(z)| < ε. en, for n > N, we deduce that From (34) and (35), we infer that en, Hence, L g : B * ω ⟶ B * ω,α,0 is actually compact. en, the assertion (III)⇒(I) is completely proved.
Next, we are clear that (II)⇒(III) and assume that L g : B * ω,0 ⟶ B * ω,α,0 is compact. Suppose that g∈B * ω,α, ln . us, we can find a specific sequence of points z n in Δ, |z n | ⟶ 1, for which where k is a positive constant and Now, assume that f n (z) � ln(1/1 − z n ). en, f n ∈ B * ω,0 with ‖f n ‖ B ω ≤ 1. Because L g is actually compact, then L g f n is also a compact concerned subset of B * ω,α,0 . In view of Lemma 3, we obtain that is is equivalent to lim |z|⟶1 − sup n f n (z) g * (z)F(ω, z) � 0. (41) Putting ε � (k/2), then we can find ρ > 0 such that sup |z|>ρ sup n f n (z) Assume that N > 0, so for n > N, |z n | > ρ, we deduce that g * z n f n z n F * ω, z n ≤ sup |z|>ρ sup n g * z n f n (z) F * ω, z n < k 2 , and this contradicts (38). Hence, the proof is established.

Hyperbolic Bloch Classes and Usual Distance
Connections between hyperbolic Bloch spaces and the metric spaces are investigated in the following emerging result. e concept of the general hyperbolic derivative is defined in [2] by Remarking that, when k � 1, we get the known usual hyperbolic derivative as given above. Now, we introduce the following concepts: e logarithmic space B * k,ω,ln is defined by B * k, ln � h * k ∈ B(Δ): ‖h‖ B * k,ω, ln � sup z∈Δ L(ω, z, ln)h * k < ∞,