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Transactions of the American Mathematical Society

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$ p$-absolutely summing operators and the representation of operators on function spaces

Author: John William Rice
Journal: Trans. Amer. Math. Soc. 188 (1974), 53-75
MSC: Primary 47B37
MathSciNet review: 0336429
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Abstract: We introduce a class of p-absolutely summing operators which we call p-extending. We show that for a logmodular function space $ A(K)$, an operator $ T:A(K) \to X$ is p-extending if and only if there exists a probability measure $ \mu $ on K such that T extends to an isometry $ T:{A^p}(K,\mu ) \to X$. We use this result to give necessary and sufficient conditions under which a bounded linear operator is isometrically equivalent to multiplication by z on a space $ {L^p}(K,\mu )$ and certain Hardy spaces $ {H^p}(K,\mu )$.

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Keywords: Banach algebra, functional calculus, p-absolutely summing operator, normal operator, subnormal operator, scalar-type spectral operator, topologically cyclic vector, Hardy space, logmodular, $ {\mathcal{L}_{p,\lambda }}$
Article copyright: © Copyright 1974 American Mathematical Society