Leibenzon’s backward shift and composition operators

Doubtsov, Evgueni

doi:10.1090/S0002-9939-01-06325-0

Leibenzon’s backward shift and composition operators
HTML articles powered by AMS MathViewer

by Evgueni Doubtsov PDF

Proc. Amer. Math. Soc. 129 (2001), 3495-3499 Request permission

Abstract:

We apply Leibenzon’s backward shift to show that the composition operator on the unit ball of $\mathbb {C}^n$ always maps the weighted Hardy space $H^2_{1-n}$ into the Hardy class $H^2$.

References

Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026
Joseph A. Cima and Peter R. Mercer, Composition operators between Bergman spaces on convex domains in $\textbf {C}^n$, J. Operator Theory 33 (1995), no. 2, 363–369. MR 1354986
G. M. Henkin, The approximation of functions in pseudo-convex domains and a theorem of Z. L. Leĭbenzon, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 37–42 (Russian, with English summary). MR 287027
Barbara D. MacCluer, Compact composition operators on $H^p(B_N)$, Michigan Math. J. 32 (1985), no. 2, 237–248. MR 783578, DOI 10.1307/mmj/1029003191
Barbara D. MacCluer and Peter R. Mercer, Composition operators between Hardy and weighted Bergman spaces on convex domains in $\textbf {C}^N$, Proc. Amer. Math. Soc. 123 (1995), no. 7, 2093–2102. MR 1254846, DOI 10.1090/S0002-9939-1995-1254846-9
Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7