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

2020/02/04 by Ghara, Soumitra, Kumar, Surjit, Pramanick, Paramita
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2002.01298

Abstract

Let Ω 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 Ω. The group \mathbb K consisting of linear transformations acts naturally on any d-tuple \boldsymbol T=(T₁,…, T_d) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that \boldsymbol T is \mathbbK-homogeneous. In this paper, we obtain a model for all \mathbbK-homogeneous d-tuple \boldsymbolT as the operators of multiplication by the coordinate functions z₁,… ,z_d 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 a bounded symmetric domain Ω of rank 2, an explicit description of the operator ∑_i=1^d T_i^*T_i is given. In general, based on this formula, we make a conjecture giving the form of this operator.

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