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Composition operators on Dirichlet-type spaces


Author: R. A. Hibschweiler
Journal: Proc. Amer. Math. Soc. 128 (2000), 3579-3586
MSC (2000): Primary 47B38; Secondary 30H05
DOI: https://doi.org/10.1090/S0002-9939-00-05886-X
Published electronically: August 17, 2000
MathSciNet review: 1778280
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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.


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  • [1] P. R. Ahern and D. N. Clark, On inner functions with $B^{p}$derivative, Mich. Math. J. 23 (1976), 107-118. MR 54:2976
  • [2] P. R. Ahern and D. N. Clark, On inner functions with $H^{p}$derivative, Mich. Math. J. 21 (1974), 115-127. MR 49:9218
  • [3] H. A. Allen and C. L. Belna, Singular inner functions with derivative in $B^{p}$, Mich. Math. J. 19 (1972), 185-188. MR 45:8844
  • [4] K. R. M. Attele, Analytic multipliers of Bergman spaces, Mich. Math. J. 31 (1984), 307-319. MR 86g:46039
  • [5] S. Axler, Multiplication operators on Bergman spaces, J. Reine Angewandt Math. 336 (1982), 26-44. MR 84b:30052
  • [6] J. A. Cima and W. R. Wogen, A Carleson measure theorem for the Bergman space on the ball, J. Operator Theory 7 (1982), 157-165. MR 83f:46022
  • [7] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. MR 97i:47056
  • [8] M. R. Cullen, Derivatives of singular inner functions, Mich. Math. J. 18 (1971), 283-287. MR 44:438
  • [9] P. Duren, Theory of $H^{p}$ Spaces, Academic Press, New York, 1970. MR 42:3552
  • [10] W. Hastings, A Carleson measure theorem for Bergman spaces, Proc. Amer. Math. Soc. 52 (1975), 237-241. MR 51:11082
  • [11] M. Jovovic and B. D. MacCluer, Composition operators on Dirichlet spaces, Acta Sci. Math. (Szeged) 63 (1997), 229-247. MR 98d:47067
  • [12] R. Kerman and E. Sawyer, Carleson measures and multipliers of Dirichlet-type spaces, Trans. Amer. Math. Soc. 309 (1988), 87-98. MR 89i:30044
  • [13] B. D. MacCluer, Compact composition operators on $H^{p}(B_{N})$, Mich. Math. J. 32 (1985), 237-248. MR 86g:47037
  • [14] B. D. MacCluer, Composition operators on $S^{p}$, Houston J. Math. 13 (1987), 245-254. MR 88h:47044
  • [15] B. D. MacCluer and J. H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Can. J. Math., Vol. 38 (1986), 878-906. MR 87h:47048
  • [16] D. J. Newman and H. S. Shapiro, The Taylor coefficients of inner functions, Mich. Math. J. 9 (1962), 249-255. MR 26:6371
  • [17] G. Piranian, Bounded functions with large circular variation, Proc. Amer. Math. Soc. 19 (1968), 1255-1257. MR 37:6464
  • [18] D. Protas, Blaschke products with derivative in $H^{p}$ and $B^{p}$, Mich. Math. J. 20 (1973), 393-396. MR 49:9217
  • [19] R. Roan, Composition operators on the space of functions with $H^{p}$-derivative, Houston J. Math. 4 (1978), 423-438. MR 58:23735
  • [20] W. Rudin, The radial variation of analytic functions, Duke Math. J. 22 (1955), 235-242. MR 18:27g
  • [21] J. H. Shapiro, Compact composition operators on spaces of boundary-regular holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 49-57. MR 88c:47059
  • [22] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993. MR 94k:47049
  • [23] J. H. Shapiro, private communication.
  • [24] J. H. Shapiro, The essential norm of a composition operator, Annals of Math. 125 (1987), 375-404. MR 88c:47058
  • [25] J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on $H^{2}$, Indiana Univ. Math. J. 23 (1973), 471-496. MR 48:4816
  • [26] D. A. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math 24 (1980), 113-139. MR 81a:30027
  • [27] N. Zorboska, Composition operators on $S_{a}$ spaces, Indiana University Math. J. 39 (1990), 847-857. MR 91k:47070
  • [28] N. Zorboska, Composition operators on weighted Dirichlet spaces, Proc. Amer. Math. Soc. 126 (1998), 2013-2023. MR 98h:47047

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Additional Information

R. A. Hibschweiler
Affiliation: Department of Mathematics, University of New Hampshire, Durham, New Hampshire 03824
Email: rah2@cisunix.unh.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05886-X
Keywords: Composition operator, Dirichlet space, Carleson measure, angular derivative
Received by editor(s): October 16, 1998
Received by editor(s) in revised form: February 12, 1999
Published electronically: August 17, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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