## Noncoincidence of the strict and strong operator topologies

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- by Joel H. Shapiro
- Proc. Amer. Math. Soc.
**35**(1972), 81-87 - DOI: https://doi.org/10.1090/S0002-9939-1972-0306878-7
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## 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.

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## Bibliographic Information

- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**35**(1972), 81-87 - MSC: Primary 46E10
- DOI: https://doi.org/10.1090/S0002-9939-1972-0306878-7
- MathSciNet review: 0306878