$$\mathbb{K}$$-homogeneous tuple of operators on bounded symmetric domains

Abstract

Let Ω be an irreducible bounded symmetric domain of rank r in ℂd. Let $$\mathbb{K}$$ be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group $$\mathbb{K}$$ 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 $$\mathbb{K}$$ -homogeneous. In this paper, we obtain a model for a certain class of $$\mathbb{K}$$ -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 $$\sum\nolimits_{i = 1}^d {T_i^\ast {T_i}} $$ is given. In general, based on this formula, we make a conjecture giving the form of this operator.