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Journal of Operator Theory

Volume 70, Issue 2, Autumn 2013  pp. 451-475.

Operator identities for standard weighted Bergman shift and Toeplitz operators

Authors:  Anders Olofsson (1) and Aron Wennman (2)
Author institution: (1) Mathematics, Faculty of Science, Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden
(2) Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden

Summary:  We prove an operator identity for the shift operator in the scale of standard weighted Bergman spaces in the unit disc. This operator identity is then applied in the context of functional calculus for the shift operator and a characterization of harmonic symbol Bergman space Toeplitz operators is obtained generalizing an earlier result by Louhichi and Olofsson. Duality arguments lead to operator inequalities and structure formulas for reproducing kernel functions which make contact with work of Richter, Shimorin, and others.

Keywords:  Bergman space, shift operator, Toeplitz operator, functional calculus, reproducing kernel function

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