Functions Preserving Sequence Spaces

Abstract

Let $A$ and $B$ be sets of real sequences. Let $F(A,B)$ denote the set of functions $f : \R \rightarrow \R$ that preserve $A$ and $B$ in the sense that $(f(a_{n})) \in B$ for all sequences $(a_{n}) \in A.$ These functions are generalizations of convergence preserving functions first introduced by Rado. We establish identities and inclusions for $F(A,B)$ when $A$ and $B$ are $l^{p}$-spaces and other well-known sequence spaces. We also characterize $F(A,B)$ in terms of elementary classes of functions. Our characterizations are motivated by the work of Bors\'{\i}k, \v{C}erve\v{n}ansk\'{y} and \v{S}al\'{a}t.