Ghara, S and Kumar, S and Pramanick, P (2022) K -homogeneous tuple of operators on bounded symmetric domains. In: Israel Journal of Mathematics, 247 (1). p. 331.
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Abstract
Let Ω be an irreducible bounded symmetric domain of rank r in ℂd. Let K be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group K consisting of linear transformations acts naturally on any d-tuple T = (T1, …, Td) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is K-homogeneous. In this paper, we obtain a model for a certain class of K-homogeneous d-tuple T as the operators of multiplication by the coordinate functions z1, …, zd on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B1(Ω). For an irreducible bounded symmetric domain Ω of rank 2, an explicit description of the operator ∑i=1dTi∗Ti is given. In general, based on this formula, we make a conjecture giving the form of this operator.
| Item Type: | Journal Article |
|---|---|
| Publication: | Israel Journal of Mathematics |
| Publisher: | Hebrew University Magnes Press |
| Additional Information: | The copyright for this article belongs to the Authors. |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 07 Jan 2022 07:02 |
| Last Modified: | 24 Jun 2022 05:16 |
| URI: | https://eprints.iisc.ac.in/id/eprint/70955 |
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