Surjective Isometries of Multiplication Operators and Weighted Composition Operators on n th Weighted-Type Banach Spaces of Analytic Functions

: This paper is to characterize isometries of multiplication operators and weighted composition operators on the n th weighted-type Banach spaces { : } n n ∈   of analytic functions on the open unit disk, of which the Bloch space and the Zygmund space are particular cases at n = 1, 2. We give characterizations of the symbols ψ and φ for which the multiplication operator M ψ and the weighted composition operator , W ψφ are surjective isometries. Moreover, we show that generalized weighted composition operators are not isometric on V n .


Introduction
Let D represent the open unit disk in the complex plane, H(D) the set of analytic functions on D, and S(D) the set of analytic self maps of D. The nth weighted-type Banach spaces { : where the norm is defined as follows: In fact, V 1 is the Bloch space, and V 2 is the Zygmund space.Recently, the iterated spaces V n have been studied in several resources, such as [1] and [2].In particular, the authors of [1] show some interesting properties of such spaces.Indeed, (V n ) is a nested sequence that is contained in the disk algebra ( ) ( ) for all n ≥ 2.Moreover, for n ∈  and , Also, for each n ≥ 3, they find that V n is an algebra.One important application of the nth weighted-type Banach spaces is in the study of approximation theory and numerical analysis.In particular, these spaces can be used to quantify the accuracy of various numerical methods for approximating functions with nth-order derivatives, such as finite difference and finite element methods.Moreover, the nth weighted-type Banach spaces can be used to establish estimates for the rates of convergence of various approximation schemes as well as to obtain error bounds for numerical solutions of differential equations.More details can be found in [3] and [4].
For ( ), H D ψ ∈ the multiplication operator on V n is the linear operator given by , for all .
the composition operator on V n is the linear operator defined by the weighted composition operator is the linear operator defined as follows: , ( ) ( ( )), for all .
Given two normed vector spaces X and Y, a linear isometry is a linear map T: X → Y, which preserves the norms,

|| || || ||
Y X Tx x = for all .x X ∈ Recall that a linear isometry is injective but not necessarily surjective.Characterizing the isometric multiplication operators and the isometric weighted composition operators on several Banach spaces is an active topic that attracts several researchers.Indeed, in 2008, Bonet et al. [5] characterized the isometric multiplication operators and the isometric weighted composition operators on the weighted Banach space .
In 2005, Colonna [8] characterized the isometric composition operators on the Bloch space, which is V 1 .This was followed by further characterizations of the isometric multiplication operators on the same space in [9] and [10].Moreover, in 2014, Li [11] studied isometries among the generalized composition operators on Bloch-type spaces.
Recently, Mas and Vukotić [12] characterized surjective isometric weighted composition operators on a general class of analytic functions, assuming such a class satisfies several axioms, but we have not been able to verify whether Axiom 4 holds for the nth weighted-type Banach spaces V n for n ≥ 2. However, we obtain the same conclusions from Theorem 2 of [12] for these spaces.
In this paper, the main goal is to expand the research carried out in [8] and [9] by characterizing the isometric composition operators and the isometric multiplication operators on the spaces V n for n ≥ 2. Furthermore, in light of what has been done in [11], we show that generalized weighted composition operators are not isometric on V n .
The following results are needed throughout this work.They provide significant properties of the spaces V n and a characterization of the invertible weighted composition operator W ψ φ on V n is bounded, then it is invertible if ψ is bounded away from zero and φ is automorphism of D. The inverse is also a weighted composition operator on V n , such that In particular, the composition operator C φ is invertible on V n if φ is an automorphism on D, and the multiplication operator M ψ on V n is invertible if ψ is bounded away from zero, such that

Surjective isometric multiplication operator on V n
In [9], the authors prove that the multiplication operators on V 1 , which is the Bloch space, are isometries if and only if the symbols are constant with modulus one.In this section, we show a similar result for all spaces V n .
The following finding is needed to express and prove the primary result in this section.Proposition 2.1.
Similar results are proven for n = 1, 2. In particular, Proof.It is clear that the multiplication operator M ψ acting on Vn is isometric if ψ is a constant of modulus 1. Suppose that the multiplication operator M ψ is isometric.By Theorem 1.2, 1/ M ψ is also isometric.Thus, by Lemma 2.2, we have that which leads to the desired conclusion.

Surjective isometric weighted composition operators on V n
The authors of [1] provide a characterization of bounded weighted composition operators on V n .Also, they provide a sufficient and necessary condition for isometric composition operators on V n for n ≥ 2. In this section, we provide a characterization for the surjective isometric weighted composition operators on V n for n ≥ 2. A comparable characterization is provided for the Bloch space V 1 in [12], as it is established that V 1 meets all the necessary axioms for a similar result to be applied.
Theorem 3.1.([1], Theorem 6.1)For n ≥ 2, the composition operator C φ on V n is isometric if φ is rotation of D.
In the following theorem, we characterize the isometric weighted composition operators on V n .Theorem 3.2.For n ≥ 2, the surjective weighted composition operator Proof.It is obvious by Theorems 2.3 and 3.1 that weighted composition operator , Conversely, suppose that , On the other hand, since φ is automorphism in Theorem 1.2, and similarly from Proposition 1.1, we get that 1 1.

ψ ∞ ≤
Hence, ψ must be a constant of modulus 1, and from Theorem 3.1, φ should be a rotation of D since W ψ,φ could be considered as an isometric composition operator when ψ is a constant of modulus 1.

Generalized weighted composition operators on V n are not isometric
The author of [11] provides characterizations of isometries among the generalized composition operators on Blochtype spaces.In this section, we show that the generalized weighted composition operators on V n are not isometric.In particular, the generalized weighted composition operators on V n are not injective.The following example illustrates this fact.
Recall that for , m ∈  the generalized weighted composition operator is defined by