A Family of Homogeneous Operators In The Cowen-Douglas Class Over The Poly-disc

Abstract

We construct a large family of positive-definite kernels $K: \mathbb{D}^n\times \mathbb{D}^n \to \mbox{M} (r, \mathbb C)$, holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup $\mbox{Möb} \times\cdots\times \mbox{Möb}$ ($n$ times) of the bi-holomorphic automorphism group of ${\mathbb{D}}^n$. The adjoint of the $n$ - tuples of multiplication operators by the co-ordinate functions on the Hilbert spaces ${\mathcal{H}}_K$ determined by $K$ is then homogeneous with respect to this subgroup. We show that these $n$ - tuples are irreducible, are in the Cowen-Douglas class ${\mathrm{B}}_r({\mathbb{D}}^n)$ and that they are mutually pairwise unitarily inequivalent.

Pages: 14