Abstract
Following the work of C. Cowen and T. Kriete on the Hardy space, we prove that under a regularity condition, all composition operators with a subnormal adjoint on A2(D) have linear fractional symbols of the form. Moreover, we show that all composition operators on the Bergman space having these symbols have a subnormal adjoint, with larger range for the parameterr than found in the Hardy space case.
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References
I. N. Baker and Ch. Pommerenke,On the iteration of analytic functions in a halfplane. II, J. London Math. Soc. (2)20 (1979), no. 2, 255–258.
J. B. Conway,The theory of subnormal operators, American Mathematical Society, Providence, RI, 1991.
C. C. Cowen,Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc.265 (1981), 69–95.
C. C. Cowen,Composition operators on H 2, J. Operator Theory9 (1983), 77–106.
C. C. Cowen,Subnormality of the Cesàro operator and a semigroup of composition operators, Indiana Univ. Math. J.33 (1984), 305–318.
C. C. Cowen,Linear fraction composition operators on H 2, Integral Equations Operator Theory11 (1988), 167–171.
Composition operators on Hilbert spaces of analytic functions: a status report, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), Amer. Math. Soc., Providence, RI, 1990, pp. 131–145.
C. C. Cowen,Transferring subnormality of adjoint composition operators, Integral Equations Operator Theory15 (1992), no. 1, 167–171.
C. C. Cowen and T. L. Kriete,Subnormality and composition operators on H 2, J. Functional Analysis81 (1988), 298–319.
C. C. Cowen and B. D. MacCluer,Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.
A. Denjoy,Sur l’iterations des fonctions analytique, C. R. Acad. Sci. Paris Sér. A182 (1926), 255–257.
M. R. Embry,A generalization of the Halmos-Bran criterion for subnormality, Acta. Sci. Math. (Szeged)35 (1973), 61–64.
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,Tables of integral transforms. Vol. I, McGraw-Hill, New York, 1954, Based, in part, on notes left by Harry Bateman.
K. Hoffman,Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
T. L. Kriete,Cosubnormal dilation semigroups on Bergman spaces, J. Operator Theory17 (1987), 191–200.
T. L. Kriete and B. D. MacCluer,Composition operators on large weighted Bergman spaces, Indiana Univ. Math. J.41 (1992), 755–788.
T. L. Kriete and H. C. Rhaly,Translation semigroups on reproducing kernel Hilbert spaces, J. Operator Theory17 (1987), 33–83.
A. Lambert,Subnormality and weighted shifts, J. London Math. Soc.14 (1976), no. 2, 476–480.
E. H. Lieb and M. Loss,Analysis, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1996.
A. E. Richman,Subnormality and composition operators on the weighted Bergman spaces, Ph.D. thesis, University of Virginia, May 2000.
W. Rudin,Functional Analysis, McGraw-Hill, New York, 1973.
W. Rudin,Real and complex analysis, third ed., McGraw-Hill, New York, 1987.
H. Sadraoui,Hyponormality of Toeplitz and composition operators, Ph.D. thesis, Purdue University, 1992.
J. Wolff,Sur l’itération des fonctions, C. R. Acad. Sci. Paris Sér. A182 (1926), 42–43, 200–201.