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

Abstract

Let $\Omega$ be an irreducible bounded symmetric domain of rank $r$ in ${\mathbb{C}}^d.$ Let $\mathbb{K}$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $\Omega$. The group $\mathbb{K}$ consisting of linear transformations acts naturally on any $d$-tuple $\mathbf{T}=(T_1,\dots ,T_d)$ of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\mathbf{T}$ is $\mathbb{K}$-homogeneous. In this paper, we obtain a model for all $\mathbb{K}$-homogeneous $d$-tuple $\mathbf{T}$ as the operators of multiplication by the coordinate functions $z_1,\dots ,z_d$ on a reproducing kernel Hilbert space of holomorphic functions defined on $\Omega$. 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 $B_1(\Omega )$. For a bounded symmetric domain $\Omega$ of rank $2$, an explicit description of the operator ${\sum }_{i=1}^d{T}_i^{\ast }{T}_i$ is given. In general, based on this formula, we make a conjecture giving the form of this operator.

17 pages