and strong operator topologies |

Joel H. Shapiro |

Proc. Amer. Math.
Soc. 35 (1972), 81--87 |

Abstract: Let E be an infinite dimensional subspace
of C(S), the space of bounded continuous functions on a localy
compact Hausdorff space S. For a regular Borel measure m on S,
each element of E can be regarded as a bounded linear operator
on L^p(mu) for 1<=p<infinity. The main result of this paper
states that the strong operator topology thus induced on E is
properly weaker than the strict topology. For E the space of
bounded analytic functions on a plane region, and m=Lebesgue
area measure on this region, this answers a question raised by
Rubel and Shields in: The space of bounded analytic functions
on a plane region (Ann. Inst. Fourier Grenoble 16 (1966),
235--277). Our methods provide information about the p-summing
properties of the of the strict topology on subspaces of C(S)
and the bounded weak-star topology on conjugate Banach spaces. |

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