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Adjoints of composition operators on Hilbert spaces of analytic functions. (English) Zbl 1108.47026

It is well-known that spectral theory naturally occupies a central place in the study of any operators acting on Hilbert spaces of holomorphic functions (including the Bergman, Hardy, and Dirichlet space). In order to get a better understanding of the spectrum of a composition operator, it is important to be able to compute its adjoint. However, it is often stated in the literature that an explicit formula for the adjoint of a composition operator exists only in some special cases and that no general formula is known. The purpose of this paper is to show that such a formula exists for every admissible symbol and in any Hilbert space of holomorphic functions with reproducing kernels. Along with some new results, all known formulas for the adjoint obtained so far follow easily as a consequence, some in an improved form.

MSC:

47B33 Linear composition operators
47B38 Linear operators on function spaces (general)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
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