Vector Valued Multiplier Spaces of Fλ-Summability, Completeness Through C0(x)-Multiplier Convergence and Continuity and Compactness of Summing Operators

Abstract

Quite recently, the authors introduced the vector valued multiplier spaces associated to the series of bounded linear operators M-f(infinity)(Sigma(k) T-k) and M-wf(infinity)(Sigma(k) T-k) by means of almost and weak almost summability, respectively; [J. Math. Anal. Appl. 484: 123651]. As was recorded as an open problem in [J. Math. Anal. Appl. 484: 123651], in this study, we introduce vector valued multiplier spaces M-f lambda(infinity)(Sigma(k) T-k) and M-wf lambda(infinity)(Sigma(k) T-k) by means of generalized almost and weak almost summability, and give a characterization of completeness of these spaces, via c(0)(X)-multiplier convergent series. We also characterize the continuity and the (weak) compactness of the summing operator S from the multiplier spaces M-f lambda(infinity)(Sigma(k) T-k) or M-wf lambda(infinity)(Sigma(k) T-k) to an arbitrary normed space Y through c(0)(X)-multiplier Cauchy and l(infinity)(X)-multiplier convergent series, respectively. Finally, we prove that if Sigma(k) T-k is l(infinity)(X)-multiplier Cauchy, then the spaces M-f lambda(infinity)(Sigma(k) T-k) and M-wf lambda(infinity)(Sigma(k) T-k) are identical. These results are more general than the corresponding consequences given in [J. Math. Anal. Appl. 484: 123651] since almost convergence can be obtained from f(lambda) -convergence under certain conditions.