A class of functions is speciied which give rise to semibounded quadratic forms on weighted Bergman spaces and thus can be interpreted as symbols of self-adjoint Berezin-Toeplitz operators. A similar class admits a probabilistic expression of the sesqui-analytic integral kernel for the associated semigroups. Both results are the consequence of a relation of Berezin-Toeplitz operators to Schrr o...

Let = G=K be a bounded symmetric domain in a complex vector space V with the Lebesgue measure dm(z) and the Bergman reproducing kernel h(z; w) ?p. Let dd (z) = h(z; z) dm(z), > ?1, be the weighted measure on. The group G acts unitarily on the space L 2 ((;) via change of variables together with a multiplier. We consider the discrete parts, also called the relative discrete series, in the irredu...

We show that the classical Szasz analytic function SN (f)(x) is related to the Bergman kernel for the Bargmann-Fock space. Then we generalize this relation to any noncompact toric Kähler manifold, defining the generalized Szasz analytic function ShN (f)(x). Then we will prove the complete asymptotic expansion of ShN (f)(x) and its scaling limit property. As examples, we will compute the general...

We shall show that the Szegő and Bergman kernels associated to a finitely connected domain in the plane are generated by only three holomorphic functions of one complex variable of the form H(z) = S(z, A) where S(z, w) is the Szegő kernel and A is a fixed point in the domain. Many other important functions of potential theory and conformal mapping theory will be shown to be rational combination...

We prove that the Bergman kernel function associated to a finitely connected planar domain can be expressed as a rational combination of two independent Ahlfors maps associated to the domain plus the derivative of one of the maps. Similar results are shown to hold for the Szegő and Poisson kernels and other objects of potential theory. These results generalize routinely to the case of a relativ...

A Correction to this paper has been published: 10.1007/s12220-020-00602-z

Let Ω = G/K be a bounded symmetric domain in a complex vector space V with the Lebesgue measure dm(z) and the Bergman reproducing kernel h(z, w). Let dμα(z) = h(z, z̄) dm(z), α > −1, be the weighted measure on Ω. The group G acts unitarily on the space L(Ω, μα) via change of variables together with a multiplier. We consider the discrete parts, also called the relative discrete series, in the irr...