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

TL;DR

This paper constructs a broad family of homogeneous operators in the Cowen-Douglas class over the polydisc using positive-definite kernels that are quasi-invariant under a subgroup of automorphisms, demonstrating their irreducibility and inequivalence.

Contribution

It introduces a new family of homogeneous operators in the Cowen-Douglas class over the polydisc via quasi-invariant kernels, expanding the understanding of such operators.

Findings

Operators are irreducible

Operators are in the Cowen-Douglas class B_r(D^n)

Operators are mutually unitarily inequivalent

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 $\mbox{M\"ob} \times\cdots\times \mbox{M\"ob}$ ($n$ 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.