Composition operators on Dirichlet-type spaces

Hibschweiler, R.

doi:10.1090/S0002-9939-00-05886-X

Composition operators on Dirichlet-type spaces
HTML articles powered by AMS MathViewer

by R. A. Hibschweiler PDF

Proc. Amer. Math. Soc. 128 (2000), 3579-3586 Request permission

Abstract:

The Dirichlet-type space $D^{p}\ (1 \leq p \leq 2$) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space $A^{p}$. Let $\Phi$ be an analytic self-map of the disc and define $C_{\Phi }(f) = f \circ \Phi$ for $f \in D^{p}$. The operator $C_{\Phi }: D^{p} \rightarrow D^{p}$ is bounded (respectively, compact) if and only if a related measure $\mu _{p}$ is Carleson (respectively, compact Carleson). If $C_{\Phi }$ is bounded (or compact) on $D^{p}$, then the same behavior holds on $D^{q}\ (1 \leq q < p$) and on the weighted Dirichlet space $D_{2-p}$. Compactness on $D^{p}$ implies that $C_{\Phi }$ is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space $D^{p}$. Inner functions which induce bounded composition operators on $D^{p}$ are discussed briefly.

References

P. R. Ahern and D. N. Clark, On inner functions with $B^{p}$ derivative, Michigan Math. J. 23 (1976), no. 2, 107–118. MR 414884, DOI 10.1307/mmj/1029001659
P. R. Ahern and D. N. Clark, On inner functions with $H^{p}$-derivative, Michigan Math. J. 21 (1974), 115–127. MR 344479, DOI 10.1307/mmj/1029001255
H. A. Allen and C. L. Belna, Singular inner functions with derivative in $B^{p}$, Michigan Math. J. 19 (1972), 185–188. MR 299796, DOI 10.1307/mmj/1029000852
K. R. M. Attele, Analytic multipliers of Bergman spaces, Michigan Math. J. 31 (1984), no. 3, 307–319. MR 767610, DOI 10.1307/mmj/1029003075
Sheldon Axler, Multiplication operators on Bergman spaces, J. Reine Angew. Math. 336 (1982), 26–44. MR 671320, DOI 10.1515/crll.1982.336.26
Joseph A. Cima and Warren R. Wogen, A Carleson measure theorem for the Bergman space on the ball, J. Operator Theory 7 (1982), no. 1, 157–165. MR 650200
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
Michael R. Cullen, Derivatives of singular inner functions, Michigan Math. J. 18 (1971), 283–287. MR 283205
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
William W. Hastings, A Carleson measure theorem for Bergman spaces, Proc. Amer. Math. Soc. 52 (1975), 237–241. MR 374886, DOI 10.1090/S0002-9939-1975-0374886-9
Mirjana Jovović and Barbara MacCluer, Composition operators on Dirichlet spaces, Acta Sci. Math. (Szeged) 63 (1997), no. 1-2, 229–247. MR 1459789
Ron Kerman and Eric Sawyer, Carleson measures and multipliers of Dirichlet-type spaces, Trans. Amer. Math. Soc. 309 (1988), no. 1, 87–98. MR 957062, DOI 10.1090/S0002-9947-1988-0957062-1
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, Composition operators on $S^p$, Houston J. Math. 13 (1987), no. 2, 245–254. MR 904956
Barbara D. MacCluer and Joel H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), no. 4, 878–906. MR 854144, DOI 10.4153/CJM-1986-043-4
D. J. Newman and Harold S. Shapiro, The Taylor coefficients of inner functions, Michigan Math. J. 9 (1962), 249–255. MR 148874, DOI 10.1307/mmj/1028998724
George Piranian, Bounded functions with large circular variation, Proc. Amer. Math. Soc. 19 (1968), 1255–1257. MR 230906, DOI 10.1090/S0002-9939-1968-0230906-X
David Protas, Blaschke products with derivative in $H^{p}$ and $B^{p}$, Michigan Math. J. 20 (1973), 393–396. MR 344478
Raymond C. Roan, Composition operators on the space of functions with $H^{p}$-derivative, Houston J. Math. 4 (1978), no. 3, 423–438. MR 512878
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
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
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
J. H. Shapiro, private communication.
Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, DOI 10.2307/1971314
J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on $H^{2}$, Indiana Univ. Math. J. 23 (1973/74), 471–496. MR 326472, DOI 10.1512/iumj.1973.23.23041
I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622. MR 81, DOI 10.1215/S0012-7094-39-00549-1
Nina Zorboska, Composition operators on $S_a$ spaces, Indiana Univ. Math. J. 39 (1990), no. 3, 847–857. MR 1078741, DOI 10.1512/iumj.1990.39.39041
Nina Zorboska, Composition operators on weighted Dirichlet spaces, Proc. Amer. Math. Soc. 126 (1998), no. 7, 2013–2023. MR 1443862, DOI 10.1090/S0002-9939-98-04266-X