Yolanda Moreno Departamento de Mateméticas Universidad de Extremadura Spain E-mail address: ymoreno@unex.es

Vector valued Sobczyk’s theorem

Abstract: We provide a vectorial version of Sobczyk’s theorem. Recall that Sobczyk’s theorem can be read as follows: every w^{∗}-null sequence of functionals (y_{n}^{∗})_{n} on a Banach space, say Y, can be extended to any space X such that X/Y is separable by a w^{∗}-null sequence of functionals. We prove a vectorial version of such result, essentially: every uniformly bounded sequence of operators T_{n} : Y → Y_{n} admitting uniformly bounded extensions to a space X such that X/Y is separable and in such a way that (T_{n}) converges to 0 in the strong operator topology (SOT) can be extended to X by a SOT -null sequence. In particular, Sobczyk’s theorem and the Rosenthal and Johnson-Oikhberg’s results related to the topic are obtained as consequence.