Compact composition operators on 𝐿¹

Shapiro, Joel H.; Sundberg, Carl

doi:10.1090/S0002-9939-1990-0994787-0

Compact composition operators on $L^ 1$
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by Joel H. Shapiro and Carl Sundberg

Proc. Amer. Math. Soc. 108 (1990), 443-449

DOI: https://doi.org/10.1090/S0002-9939-1990-0994787-0

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Abstract:

The composition operator induced by a holomorphic self-map of the unit disc is compact on ${L^1}$ of the unit circle if and only if it is compact on the Hardy space ${H^2}$ of the disc. This answers a question posed by Donald Sarason: it proves that Sarason’s integral condition characterizing compactness on ${L^1}$ is equivalent to the asymptotic condition on the Nevanlinna counting function which characterizes compactness on ${H^2}$.

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