A family of homogeneous operators in the Cowen–Douglas class over the poly-disc
Volume 271 / 2023
Studia Mathematica 271 (2023), 65-84
MSC: Primary 47B13; Secondary 47B32, 20C25, 53C07.
DOI: 10.4064/sm220630-10-1
Published online: 13 March 2023
Abstract
We construct a large family of positive definite kernels $K: \mathbb D^n\times \mathbb D^n \to \mathrm M (r, \mathbb C)$, holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup Möb$\,\times \cdots \times\,$Möb ($n$ times) of the bi-holomorphic automorphism group of $\mathbb D^n$. The adjoint of the $n$-tuple of the multiplication operators by the co-ordinate functions is then homogeneous with respect to this subgroup on the Hilbert space $\mathcal H_K$ determined by $K$. We show that these $n$-tuples are irreducible, are in the Cowen–Douglas class $\mathrm B_r(\mathbb D^n)$ and are mutually pairwise unitarily inequivalent.
