Hideyuki Ishi

Korányi's lemma for homogeneous Siegel domains of type II. Applications and extended results

We show that the modulus of the Bergman kernel B(z, ζ) of a general homogeneous Siegel domain of type II is ”almost constant” uniformly with respect to z when ζ varies inside a Bergman ball. The control is expressed in terms of the Bergman distance. This result was proved by A. Koranyi for symmetric Siegel domains of type II. Subsequently, R. R. Coifman and R. Rochberg used this result to establish an atomic decomposition theorem and an interpolation theorem by functions in Bergman spaces \(A^p\) on these domains. The atomic decomposition theorem and the interpolation theorem are extended here to the general homogeneous case using the same tools. We further extend the range of exponents \(p\) via functional analysis using recent estimates. DOI: 10.1017/S0004972714000033

Matrix Realization of a Homogeneous Cone

Based on the theory of compact normal left-symmetric algebra (clan), we realize every homogeneous cone as a set of positive definite real symmetric matrices, where homogeneous Hessian metrics as well as a transitive group action on the cone are described efficiently.

The unitary representations parametrized by the Wallach set for a homogeneous bounded domain

Abstract. We show that the unitary representations corresponding to the parameter c running through the Wallach set for a homogeneous bounded domain are mutually inequivalent.

Explicit Formula of Koszul–Vinberg Characteristic Functions for a Wide Class of Regular Convex Cones

The Koszul–Vinberg characteristic function plays a fundamental role in the theory of convex cones. We give an explicit description of the function and related integral formulas for a new class of convex cones, including homogeneous cones and cones associated with chordal (decomposable) graphs appearing in statistics. Furthermore, we discuss an application to maximum likelihood estimation for a certain exponential family over a cone of this class.

Unitary holomorphic multiplier representations over a homogeneous bounded domain

Abstract We consider the unitarizability of multiplier representations of transformation groups defined on Hilbert spaces of holomorphic functions on a homogeneous bounded domain. In particular, for the Iwasawa subgroup of the holomorphic automorphism group the classification of the unitary multiplier representations is accomplished by making use of results in [Ishi, J. Funct. Anal. 167: 425–462, 1999]. As an application, the Wallach set of the homogeneous bounded domain is described.