Compact, nuclear, and Hilbert-Schmidt composition operators on H^2


Joel H. Shapiro and Peter D. Taylor


Indiana Univ. Math. J. 23 (1973/74) , 471--496
Abstract: My first paper on composition operators. It characterizes the Hilbert-Schmidt composition operators and shows that non-finiteness of the angular derivative is necessary for compactness. A sufficient condition for compactness involving the angular derivative---much strengthened in later papers---is proved, and these results are used to give examples of compact and noncompact composition operators. In particular, if the inducing map takes the unit disc into a polygon inscribed in the unit circle, then the induced composition operator is in every Schatten class.
Download .pdf file (880KB)