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

Rice, John William

doi:10.1090/S0002-9947-1974-0336429-9

$p$-absolutely summing operators and the representation of operators on function spaces
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Trans. Amer. Math. Soc. 188 (1974), 53-75 Request permission

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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