Homogeneous operators on Hilbert spaces of holomorphic functions |
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Authors: | Adam Korá nyi |
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Affiliation: | a Lehman College, The City University of New York, Bronx, NY 10468, USA b Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India |
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Abstract: | In this paper we construct a large class of multiplication operators on reproducing kernel Hilbert spaces which are homogeneous with respect to the action of the Möbius group consisting of bi-holomorphic automorphisms of the unit disc D. Indeed, this class consists of exactly those operators for which the associated unitary representation of the universal covering group of the Möbius group is multiplicity free. For every m∈N we have a family of operators depending on m+1 positive real parameters. The kernel function is calculated explicitly. It is proved that each of these operators is bounded, lies in the Cowen-Douglas class of D and is irreducible. These operators are shown to be mutually pairwise unitarily inequivalent. |
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Keywords: | Homogeneous operators Homogeneous holomorphic Hermitian vector boundle Associated representation Cowen-Douglas class Reproducing kernel function |
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