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Schramm spaces and composition operators

Wróbel Małgorzata

Journal of Applied Mathematics and Computational Mechanics

2023

Vol. 22, nr 2

87--98

We give some properties of Schramm functions; among others, we prove that the family of all continuous piecewise linear functions defined on a real interval I is contained in the space ΦBV (I) of functions of bounded variation in the sense of Schramm. Moreover, we show that the generating function of the corresponding Nemytskij composition operator acting between Banach spaces CΦBV (I) of continuous functions of bounded Schramm variation has to be continuous and additionally we show that a space CΦBV (I) has the Matkowski property.

On the analytic α-Lipschitz vector-valued operators

Shokri Abbasali

Journal of Mathematics and Applications

2022

Vol. 45

181--190

Let (X, d) be a non-empty compact metric space in C, (B, ∥ . ∥) be a commutative unital Banach algebra over the scalar field F(= R or C) and α ∈ R with 0 < α ≤ 1. In this work, first we define the analytic α-Lipschitz B-valued operators on X and denote the Banach algebra of all these operators by Lipα A(X, B). When B = F, we write Lipα A(X) instead of Lipα A(X, B). Then we study some interesting results about Lipα A(X, B), including the relationship between Lipα A(X, B) with Lipα A(X) and B, and also characterize the characters on Lipα A(X, B).