New constructions of homogeneous operators

New examples of homogeneous operators involving infinitely many parameters are constructed. They are realized on Hilbert spaces of holomorphic functions with reproducing kernels which are computed explicitly. All the examples are irreducible and belong to the Cowen Douglas class. Even though the construction is completely explicit, it is based on certain facts about Hermitian holomorphic homogeneous vector bundles. These facts also make possible a description of all homogeneous Cowen Douglas operators, in a somewhat less explicit way. Ou construit une nouvelle famille d’examples d’opérateurs homogènes dépendant d’une infinité de paramètres. Les exemples sont réalisés sur des espaces de fonctions holomorphes possédant des noyaux reproduisants qu’on calcule explicitement. Les exemples sont tous des opérateurs irréductibles appartenant à la classe de Cowen Douglas. Tout en étant complètement explicite, la construction est fondée sur certaines propriétés des fibrés vectoriels hermitiens holomorphes homogènes. Ces propriétés permettent aussi une description, un peu moins explicite, de tous les opérateurs homogènes de la classe de Cowen Douglas. An operator T on a Hilbert space is said to be homogeneous if its spectrum is contained in the closure of the unit disc D in C and if g(T ) is unitarily equivalent to T for every element g of the holomorphic automorphism group G of D. There are general results about such operators, but relatively few examples are known (cf. [7,2,3,1]). In this note a large family of examples is constructed and a step is made towards the description of all such operators in the Cowen Douglas class of D (see [4]). 1. Construction of the examples Let G̃ denote the universal covering group of G. For λ > 0, let A denote the Hilbert space of holomorphic functions on D with reproducing kernel (1− zw̄)−2λ. The well-known discrete series D λ of unitary representation of G̃ acts on A by Email addresses: [email protected] (Adam Korányi), [email protected] (Gadadhar Misra). Preprint submitted to Elsevier Science 14th March 2006 (D λ (g)f)(z) = (∂(g−1z) ∂z )λ f(g−1(z)) with the power defined to ensure continuity on G̃× D. Let m ≥ 1 be an integer and let λ > m2 . For 0 ≤ j ≤ m we write λj = λ− m 2 +j and define the operator Γj : Aj→ Hol(D,C) by (Γjf)` =  0 if ` < j ( ` j ) 1 (2λj) `−j f (`−j) if ` ≥ j . Denote by Aj the image of Γj . The algebraic sum of the spaces Aj can be shown to be direct. Hence, for any μ1, . . . , μm and μ0 = 1 we can define a norm on the direct sum by 〈 m ∑