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 homogeneous with respect to the action of the Mö bius group consisting of bi-holomorphic automorphisms of the unit disc D. For every m ∈ 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 D and is irreducible. These operators are shown to be mutually pairwise unitarily inequivalent.