Composition operators on analytic Lipschitz spaces

Madigan, Kevin M.

doi:10.1090/S0002-9939-1993-1152987-6

Composition operators on analytic Lipschitz spaces
HTML articles powered by AMS MathViewer

by Kevin M. Madigan PDF

Proc. Amer. Math. Soc. 119 (1993), 465-473 Request permission

Abstract:

If $X$ is a Banach space of functions analytic on the disk and $\varphi :{\mathbf {D}} \to {\mathbf {D}}$ is analytic, one can define the composition operator ${C_\varphi }$ on $X$ by ${C_\varphi }f: = f \circ \varphi$. This paper discusses the boundedness and $w$-compactness of composition operators on the analytic Lipschitz spaces ${\mathcal {A}_\alpha },\;0 < \alpha < 1$.

References

Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
P. L. Duren, B. W. Romberg, and A. L. Shields, Linear functionals on $H^{p}$ spaces with $0<p<1$, J. Reine Angew. Math. 238 (1969), 32–60. MR 259579
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Raymond C. Roan, Composition operators on a space of Lipschitz functions, Rocky Mountain J. Math. 10 (1980), no. 2, 371–379. MR 575309, DOI 10.1216/RMJ-1980-10-2-371
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Joel H. Shapiro, Compact composition operators on spaces of boundary-regular holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), no. 1, 49–57. MR 883400, DOI 10.1090/S0002-9939-1987-0883400-9
Ke He Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23 (1993), no. 3, 1143–1177. MR 1245472, DOI 10.1216/rmjm/1181072549

Distances and Banach spaces of holomorphic functions on complex domains