Compact composition operators on \(L^ 1\). (English) Zbl 0704.47018

Shapiro, Joel H.; Sundberg, Carl

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

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