Homogeneous operators on Hilbert spaces of holomorphic functions

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 Möbius group consisting of bi-holomorphic automorphisms of the unit disc D . Indeed, this class consists of exactly those operators for which the associated unitary representation of the universal covering group of the Möbius group is multiplicity free. 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.