Noncoincidence of the Strict and Strong Operator Topologies

Abstract

Let E be an infinite-dimensional linear subspace of $C(S)$, the space of bounded continuous functions on a locally compact Hausdorff space S. If $\mu$ is a regular Borel measure on S, then each element of E may be regarded as a multiplication operator on ${L^p}(\mu )(1 \leqq p < \infty )$. Our main result is that the strong operator topology this identification induces on E is properly weaker than the strict topology. For E the space of bounded analytic functions on a plane region G, and $\mu$ Lebesgue measure on G, this answers negatively a question raised by Rubel and Shields in [9]. In addition, our methods provide information about the absolutely p-summing properties of the strict topology on subspaces of $C(S)$, and the bounded weak star topology on conjugate Banach spaces.