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.

Homogeneous Operators, Jet Construction and Similarity

“…We briefly describe the class of homogeneous operators which appear in [14]. The construction of these homogeneous operators can be thought of as "another jet construction".…”

“…The kernel B (λ,μ) is positive definite. Indeed, it is the reproducing kernel for the Hilbert space A (λ,μ) (D) of C m+1 -valued holomorphic functions described in [14]. The Hermitian holomorphic vector bundle associated with B (λ,μ) is denoted by E (λ,μ) .…”

“…The operators in H 1,m+1 (D) are irreducible and homogeneous such that the associated representation is multiplicity-free [14,15]. Although, it is not clear at the outset that there exist (α, β) and (λ, μ) such that the two homogeneous operators M (α,β) m and M (λ,μ) are unitarily equivalent.…”

“…The multiplication operators constructed via the first jet construction are known as the "Generalized Wilkins' operators" [3, p. 428]. However, there is an easy modification of the first jet construction that allows us to construct the entire family of homogeneous operators which were first exhibited in [14].…”

“…Unfortunately, this construction does not give a complete list of homogeneous operators in B n (D) except in the case of n = 1, 2. Nevertheless, using a very simple invertible transformation, we enlarge our list to match with the large family of homogeneous operators H 1,n (D) constructed in [14,15]. The class H 1,n (D) provides a complete list of those irreducible homogeneous operators in B n (D) whose associated representations are multiplicityfree.…”