On operator valued sequences of multipliers and R-boundedness

Abstract: In recent papers (cf. [J.L. Arregui, O. Blasco, -Summing sequences, J. Math. Anal. Appl. 274 (2002) 812–827; J.L. Arregui, O. Blasco, -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 -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 -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 and be two Banach spaces whose elements are sequences of vectors in X and Y, respectively, and which contain the spaces and of all X-valued and Y-valued sequences which are eventually zero, respectively. Generally spoken, a sequence of bounded linear operators is called a multiplier sequence from to if the linear operator from into which maps onto is bounded with respect to the norms on and , respectively. Several cases where and are different (classical) spaces of sequences, including, for instance, the spaces 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. [Copyright &y& Elsevier]