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Toeplitz Operators on Bergman Spaces

Published online by Cambridge University Press:  20 November 2018

Sheldon Axler ,
John B. Conway  and
Gerard McDonald
Sheldon Axler
Affiliation:
Michigan State University, East Lansing, Michigan
John B. Conway
Affiliation:
Indiana University, Bloomington, Indiana
Gerard McDonald
Affiliation:
Western Illinois University, Macomb, Illinois
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Let G be a bounded, open, connected, non-empty subset of the complex plane C. We put the usual two dimensional (Lebesgue) area measure on G and consider the Hilbert space L2(G) that consists of the complex-valued, measurable functions defined on G that are square integrable. The inner product on L2(G) is given by the norm ‖h‖2 of a function h in L2(G) is given by ‖h‖2 = (∫G|h|2)1/2.

The Bergman space of G, denoted La2(G), is the set of functions in L2(G) that are analytic on G. The Bergman space La2(G) is actually a closed subspace of L2(G) (see [12 , Section 1.4]) and thus it is a Hilbert space.

Let G denote the closure of G and let C(G) denote the set of continuous, complex-valued functions defined on G.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 2 , 01 April 1982 , pp. 466 - 483
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Adams, D. R. and Polking, J. C., The equivalence of two definitions of capacity, Proceedings A.M.S. 37 (1973), 529–534.Google Scholar
2. Ahlfors, L. V., Conformai invariants; Topics in geometric function theory (McGraw- Hill, 1973).Google Scholar
3. Ahlfors, L. and Beurling, A., Conformai invariants and function-theoretic null sets, Acta Math. (Uppsala) 83 (1950), 101–129.Google Scholar
4. Arveson, W., An invitation to C*-algebras, Springer-Verlag, Graduate Texts in Mathematics, Volume 39 (1976).Google Scholar
5. Berger, C. A. and Shaw, B. I., Self commutator s of multicyclic hyponormal operators are always trace class, Bulletin A.M.S. 79 (1973), 1193–1199.Google Scholar
6. Berger, C. A. and Shaw, B. I., Intertwining, analytic structure, and the trace norm estimate, in Proceedings of a conference on operator theory, Springer-Verlag, Lecture Notes in Mathematics, Volume 346 (1973).Google Scholar
7. Berger, C. A. and Shaw, B. I., Hyponormality: its analytic consequences, preprint.Google Scholar
8. Bergman, S., The kernel function and conformai mapping, American Mathematical Society, Mathematical Surveys, Number V, Second Edition (1970).Google Scholar
9. Carleson, L., Selected problems on exceptional sets (D. Van Nostrand, 1967).Google Scholar
10. Coburn, L. A., Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973), 433–439.Google Scholar
11. Douglas, R. G., Banach algebra techniques in operator theory (Academic Press, 1972).Google Scholar
12. Epstein, B., Orthogonal families of analytic functions (Macmillan, 1965).Google Scholar
13. Gamelin, T. W., Uniform algebras (Prentice-Hall, 1969).Google Scholar
14. Garnett, J., Analytic capacity and measure, Springer Verlag, Lecture Notes in Mathematics, Volume 297 (1972).Google Scholar
15. Goldberg, S., Unbounded linear operators (McGraw-Hill, 1966).Google Scholar
16. Harvey, R. and Polking, J. C., A notion of capacity which characterizes removable singularities, Trans. A.M.S. 169 (1972), 183–195.Google Scholar
17. Hedberg, L. I., Approximation in the mean by analytic functions, Trans. A.M.S. 163 (1972), 157–171.Google Scholar
18. Jewell, N. P., Toeplitz operators on the Bergman spaces and in several complex variables, Proc. London Math. Soc. 41 (1980), 193–216.Google Scholar
19. Keough, G. E., Subnormal operators. Toeplitz operators, and spectral inclusion, Trans. A.M.S. 263 (1981), 125–135.Google Scholar
20. McDonald, G., Fredholm properties of a class of Toeplitz operators on the ball, Indiana Univ. Math. J. 26 (1977), 567–576.Google Scholar
21. McDonald, G. and Sundberg, C., Toeplitz operators on the disc, Indiana Univ. Math. J. 28 (1979), 595–611.Google Scholar
22. Pearcy, C. M., Some recent developments in operator theory, Conference board of the mathematical sciences, Regional conference series in mathematics 36 (1978).Google Scholar
23. Shields, A. L. and Wallen, L. J., The commutants of certain Hilbert space operators, Indiana Univ. Math. J. 20 (1971), 777–788.Google Scholar