(E,F)-multipliers and applications

TL;DR

This paper investigates the structure of $(E,F)$-multiplier spaces between symmetric sequence spaces, establishing embeddings into classical $(p,q)$-multiplier spaces and providing examples that extend classical Banach space theory results.

Contribution

It introduces new results on embeddings of $(E,F)$-multiplier spaces into classical spaces and constructs examples of symmetric sequence spaces with complex tensor product properties.

Findings

Continuous embedding of $(E,F)$-multiplier spaces into classical $(p,q)$-multiplier spaces.

Examples of symmetric sequence spaces with non-isomorphic tensor products.

Extension of classical results on Banach space subspaces with unconditional bases.

Abstract

For two given symmetric sequence spaces $E$ and $F$ we study the $(E,F)$-multiplier space, that is the space all of matrices $M$ for which the Schur product $M\ast A$ maps $E$ into $F$ boundedly whenever $A$ does. We obtain several results asserting continuous embedding of $(E,F)$-multiplier space into the classical $(p,q)$-multiplier space (that is when $E=l_p$, $F=l_q$). Furthermore, we present many examples of symmetric sequence spaces $E$ and $F$ whose projective and injective tensor products are not isomorphic to any subspace of a Banach space with an unconditional basis, extending classical results of S. Kwapie\'{n} and A. Pe{\l}czy\'{n}ski and of G. Bennett for the case when $E=l_p$, $F=l_q$.