Homogeneous operators on Hilbert spaces of holomorphic functions -- I

Abstract

In this paper we construct a large class of multiplication operators on reproducing kernel Hilbert spaces which are {\em homogeneous} with respect to the action of the Möbius group consisting of bi-holomorphic automorphisms of the unit disc $\mathbb{D}$. For every $m\in \mathbb{N}$ we have a family of operators depending on $m+1$ positive real parameters. The kernel function is calculated explicitly. It is proved that each of these operators is bounded, lies in the Cowen-Douglas class of $\mathbb{D}$ and is irreducible. These operators are shown to be mutually pairwise unitarily inequivalent.