Berezin Transform on Real Bounded Symmetric Domains

Let D be a bounded symmetric domain in a complex vector space VC with a real form V and D = D∩V = G/K be the real bounded symmetric domain in the real vector space V . We construct the Berezin kernel and consider the Berezin transform on the L2-space on D. The corresponding representation of G is then unitarily equivalent to the restriction to G of a scalar holomorphic discrete series of holomorphic functions on D and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the L2-space. Introduction The main purpose of the present paper is to calculate the spectral symbol of the Berezin transform on real bounded symmetric domains. To explain our results and motivations we let D be the unit disk in the complex plane with the Lebesgue measure dm(z). We consider the weighted Bergman space H (ν > 1) of holomorphic functions on D square integrable with respect to the measure (1 − |z|2)ν−2dm(z). It has up to some positive constant the reproducing kernel Kw(z) = K(z, w) = (1 − zw̄)−ν . Moreover the group Gc = SU(1, 1) of fractional transformations of D acts on the space H via f(z) 7→ f(gz)g′(z) ν2 and it forms an irreducible unitary (projective) representation. Consider the subgroup SO(1, 1) consisting of transformations of the form z 7→ az+b bz+a with a, b ∈ R and a − b = 1. Thus it is of interest to study the irreducible decomposition of the weighted Bergman space under the subgroup G = SO(1, 1). For that purpose we consider the unit interval D = D ∩ R = (−1, 1) as a trivial symmetric space G/K = SO(1, 1)/{±1} and the restriction of holomorphic functions in H to the interval D. More precisely, consider R : H → C∞(D), Rf(x) = f(x)(1 − x) ν2 , x ∈ D. Let L(D, dμ0) be the L space on D with the SO(1, 1)-invariant measure dμ0(x) = dx (1−x2) , whose decomposition under SO(1, 1) can be done via the Mellin transform (see below). The restriction R is a bounded operator from H into the space L(D, dμ0) with dense image, and intertwines the respective actions of SO(1, 1), Received by the editors January 16, 2000 and, in revised form, October 10, 2000. 2000 Mathematics Subject Classification. Primary 22E46, 43A85, 32M15, 53C35.