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 M\"{o}bius group consisting of bi-holomorphic automorphisms of the unit disc $\mathbb D$. For every $m \in N$ 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.

This paper presents a large class of homogeneous multiplication operators on reproducing kernel Hilbert spaces, proving that they are bounded, irreducible, and mutually pairwise unitarily inequivalent.

Homogeneous operators on Hilbert spaces of holomorphic functions-I

Key Takeaway: This paper presents a large class of homogeneous multiplication operators on reproducing kernel Hilbert spaces, proving that they are bounded, irreducible, and mutually pairwise unitarily inequivalent.