Homogeneous tuples of operators and holomorphic discrete series representation of some classical groups

Misra, Gadadhar ; Narasimha Sastry, N. S. (1990) Homogeneous tuples of operators and holomorphic discrete series representation of some classical groups Journal of Operator Theory, 24 (1). pp. 23-32. ISSN 0379-4024

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Abstract

Let T = (T1....., Tn) be a n-tuple of bounded linear operators on a fixed Hilbert space . H and let φ be a biholomorphic automorphism of Ω, the joint spectrum of T. In this paper, we consider those n-tuples T for which the joint spectrum Ω is of the form G/K, a bounded symmetric domain. Let φ be any biholornorphic automorphism of the domain Ω. Define, phi(T) via a suit­ able functional calculus and call a n-tuple of operators T homogeneous if φ(T) .is simultaneously unitarily equivalent to T for every automorphism φ of Ω. For each homogeneous operator T, let Uφ be a unitary] operator implimenting this equivalence. We obtain a characterisation of all the homogeneous operators Cowen-Douglas class and show that it is possible to choose the unitary Uφ in such a way that the map φ→ Uφ -1 is a unitary representation of the group of of biholomorpic automorphisms of Ω.

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