On operator valued sequences of multipliers and R-boundedness

Abstract

In recent papers (cf. [J.L. Arregui, O. Blasco, ( p , q ) -Summing sequences, J. Math. Anal. Appl. 274 (2002) 812–827; J.L. Arregui, O. Blasco, ( p , q ) -Summing sequences of operators, Quaest. Math. 26 (2003) 441–452; S. Aywa, J.H. Fourie, On summing multipliers and applications, J. Math. Anal. Appl. 253 (2001) 166–186; J.H. Fourie, I. Röntgen, Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2) (2003) 629–644]) the concept of ( p , q ) -summing multiplier was considered in both general and special context. It has been shown that some geometric properties of Banach spaces and some classical theorems can be described using spaces of ( p , q ) -summing multipliers. The present paper is a continuation of this study, whereby multiplier spaces for some classical Banach spaces are considered. The scope of this research is also broadened, by studying other classes of summing multipliers. Let E ( X ) and F ( Y ) be two Banach spaces whose elements are sequences of vectors in X and Y, respectively, and which contain the spaces c 00 ( X ) and c 00 ( Y ) of all X-valued and Y-valued sequences which are eventually zero, respectively. Generally spoken, a sequence of bounded linear operators ( u n ) ⊂ L ( X , Y ) is called a multiplier sequence from E ( X ) to F ( Y ) if the linear operator from c 00 ( X ) into c 00 ( Y ) which maps ( x i ) ∈ c 00 ( X ) onto ( u n x n ) ∈ c 00 ( Y ) is bounded with respect to the norms on E ( X ) and F ( Y ) , respectively. Several cases where E ( X ) and F ( Y ) are different (classical) spaces of sequences, including, for instance, the spaces Rad ( X ) of almost unconditionally summable sequences in X, are considered. Several examples, properties and relations among spaces of summing multipliers are discussed. Important concepts like R-bounded, semi- R-bounded and weak- R-bounded from recent papers are also considered in this context.