Shapiro, Joel H.; Sundberg, Carl Compact composition operators on \(L^ 1\). (English) Zbl 0704.47018 Proc. Am. Math. Soc. 108, No. 2, 443-449 (1990). Composition of a holomorphic function from the unit disc to itself with a Poisson integral induces a composition operator on \(L^ 1\) of the unit circle. D. Sarason recently proved that the compactness of such an operator on \(L^ 1\) implies its compactness on the Hardy space \(H^ 2\). The present paper establishes the converse. Reviewer: E.Azoff Cited in 16 Documents MSC: 47B07 Linear operators defined by compactness properties 47B38 Linear operators on function spaces (general) 30D55 \(H^p\)-classes (MSC2000) Keywords:compact composition operators on \(L^ 1\); Nevanlinna counting function; Riesz mass; Poisson integral; Hardy space PDF BibTeX XML Cite \textit{J. H. Shapiro} and \textit{C. Sundberg}, Proc. Am. Math. Soc. 108, No. 2, 443--449 (1990; Zbl 0704.47018) Full Text: DOI