Siegel Domain Research Articles

Holomorphic mappings of domains in operator spaces

Our object is to give an overview of some basic results about holomorphic mappings of circular domains in various spaces of operators. We begin by considering C*-algebras and pass to J*-algebras and other spaces when this seems natural. Our first result is a simple extension of the maximum principle where the unitary operators play the role of the unit circle. We illustrate the power of this result by deducing some classical theorems of functional analysis in a straightforward way. Next we apply Cartan’s uniqueness theorem to determine biholomorphic mappings and to show that linear mappings between certain operator domains are Jordan isomorphisms. This motivates the discussion of homogeneous domains. Our first examples of homogeneous domains are the open unit balls of J*-algebras (which include all the classical domains.) Next we discuss some affinely homogeneous upper half-planes in spaces of operators called operator Siegel domains of genus 2. Although these are always holomorphically equivalent to a bounded domain, they are (as far as I know) not necessarily holomorphically equivalent to a ball. Our last examples are the domains of linear fractional transformations. These are symmetric affinely homogeneous domains which are never holomorphically equivalent to a bounded domain. Finally we show how certain extensions of the Riemann removable singularity theorem allow us to determine the automorphisms of domains where an indefinite operator-valued form is positive. This includes the operator analogue of the exterior of the open unit disc.

Ramanujan's Master Theorem for Hermitian Symmetric Spaces

In this paper we generalize Ramanujan's Master Theorem to the context of a Siegel domain of Type II, which is the Hermitian symmetric space of a real non-compact semisimple Lie group of finite center. That is, the spherical transform of a spherical series can be expressed in terms of the coefficients of this series.

Barrier Functions in Interior Point Methods

We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone. This provides a simple method for calculating the universal barrier for homogeneous cones. We identify some known barriers as the universal barrier scaled by an appropriate constant. We also calculate some new universal barrier functions. Our results connect the field of interior point methods to several branches of mathematics such as Lie groups, Jordan algebras, Siegel domains, differential geometry, complex analysis of several variables, etc.

On the automorphisms of Siegel domains

We prove that every holomorphic automorphism of a strongly pseudoconvex Siegel domain with no pole on the edge is an affine automorphism. Some related results are also given.

The polynomial topological complexity of Fatou-Julia sets

We analyze the polynomial topological complexity of Fatou-Julia sets, and show that for Sullivan domains of such sets the complexity is equal to 1 in the case of attracting, superattracting and Siegel domains, while for parabolic domains it is greater than 1.

Pseudohermitian geometry and interpolation manifolds

We prove a vanishing theorem for interpolation manifolds in the boundary of a Siegel domain and characterize the smooth submanifolds of CPn which arise as projections via of interpolation manifolds in the boundary of a complete circular domain.

Admissible convergence for the Poisson-Szegö integrals

We prove almost everywhere semirestricted admissible convergence of the Poisson-Szego integrals ofL p functions (1 <p ≤ ∞) on the Bergman-Shilov boundary of a Siegel domain. In the case of symmetric domains our theorem can be deduced from the results by Peter Sjogren on admissible convergence to the boundary of Poisson integrals on symmetric spaces, although semirestricted admissible convergence means here a more general approach to the boundary then originally defined for symmetric spaces.

Estimates for the Poisson kernels and a Fatou type theorem applications to analysis on Siegel domains

This is a short description of some results obtained by Ewa Damek, Andrzej Hulanicki, Richard Penney and Jacek Zienkiewicz. They belong to harmonic analysis on a class of solvable Lie groups called NA. We apply our results to analysis on classical Siegel

ON THE THEORY OF THE MATRIX RICCATI EQUATION. II

This paper is a continuation of the author's previous paper [1] with the same title, which contains an investigation of properties of the matrix Riccati equation with variable coefficients that arises from variational problems. For the investigation the matrix cross-ratio was introduced - an invariant of a quadruple of points in the Grassmann manifold of -dimensional subspaces in a -dimensional linear space with respect to the action of the unimodular group of generalized linear fractional transformations on the Grassmann manifold. The properties of the matrix cross-ratio are investigated in the present article; classes of complex Riccati equations giving rise to flows on homogeneous Siegel domains of types I, II, and III are distinguished; a matrix analogue of the Schwarzian derivative is introduced. Bibliography: 7 titles.

Bessel Functions on Quaternionic Siegel Domains

This paper applies the theory of operator-valued Bessel functions on Schrödinger-Fock spaces and Siegel domains of type II developed in (H. Ding, J. Funct. Anal. 106 (1992), 189-225) to harmonic analysis on quaternionic Hermitian symmetric spaces and the representation theory of the groups O *(2 m) for m odd. Using the reduced Bessel function K π we study the harmonic representation and holomorphic discrete series for these groups.

Integral Representations for Some Classes of Functions Holomorphic in a Siegel Domain

Integral Representations for Some Classes of Functions Holomorphic in a Siegel Domain

Siegel domain realization of pseudo-Hermitian symmetric manifolds

For some pseudo-Hermitian symmetric manifolds there are Zariski open subsets which are biholomorphically equivalent to tube domains. These tube domains correspond to homogeneous nonconvex cones.

Lp-Fourier transforms on nilpotent Lie groups and solvable Lie groups acting on Siegel domains

<i>L</i>^<i>p</i>-Fourier transforms on nilpotent Lie groups and solvable Lie groups acting on Siegel domains

Asymptotic behavior of the curvature of the Bergman metric of the thin domains

The Riemann sectional curvature tensor of the Bergman metric of the domains that are thin intersections of the (strongly) pseudoconvex domains in C n with C 2 boundaries, n ≥ 2, is asymptotically equivalent to that of corresponding Siegel domains

Operator-valued Bessel functions, holomorphic discrete series, and harmonic representations of U(p, q)

This paper presents representation-theoretic applications of the theory of operator-valued Bessel functions, which was developed in the first paper of this series. Using the reduced Bessel function K π we study the harmonic representation and holomorphic discrete series for the group U(p,q) of biholomorphic automorphisms of the Siegel domains of type II.

Harmonic analysis on solvable extensions of H-type groups

To each groupN of Heisenberg type one can associate a generalized Siegel domain, which for specialN is a symmetric space. This domain can be viewed as a solvable extensionS =NA ofN endowed with a natural left-invariant Riemannian metric. We prove that the functions onS that depend only on the distance from the identity form a commutative convolution algebra. This makesS an example of a harmonic manifold, not necessarily symmetric. In order to study this convolution algebra, we introduce the notion of “averaging projector” and of the corresponding spherical functions in a more general context. We finally determine the spherical functions for the groupsS and their Martin boundary.

Operator-valued Bessel functions on Schrödinger-Fock spaces and Siegel domains of type II

In this paper, a general theory of operator-valued Bessel functions on mixed Schrödinger-Fock spaces and Siegel domains of type II is presented. In general, Bessel functions J λ are associated to the partial Fourier transform on a mixed Schrödinger-Fock space. J λ can be realized as a reduced Bessel function K π , which is a generalized inverse Laplace transform on a Siegel domain of type II. The application of this theory to representation theory will be studied in the next paper.

L2-spaces and Sub-Laplacians on homogeneous groups

In this paper, we introduce a special class of nilpotent Lie groups defined by hermitian maps, which includes all the groups of affine holomorphic automorphisims of Siegel domains of type II, in particular, the Heisenberg group. And we study harmonic analysis on these groups as spectral theory of the associated Sub-Laplacian instead of the group representation theory in usual way.

ON THE THEORY OF THE MATRIX RICCATI EQUATION

This paper is a continuation of the author's previous paper [1] with the same title, which contains an investigation of properties of the matrix Riccati equation with variable coefficients that arises from variational problems. For the investigation the matrix cross-ratio was introduced - an invariant of a quadruple of points in the Grassmann manifold of -dimensional subspaces in a -dimensional linear space with respect to the action of the unimodular group of generalized linear fractional transformations on the Grassmann manifold. The properties of the matrix cross-ratio are investigated in the present article; classes of complex Riccati equations giving rise to flows on homogeneous Siegel domains of types I, II, and III are distinguished; a matrix analogue of the Schwarzian derivative is introduced.Bibliography: 7 titles.

Complex lie semigroups, hardy spaces and the gelfand-gindikin program

Let G be the group of motions of a bounded symmetric domain and let G C be its complexification. It is known that there exist open subsemigroups Γ ⊂ G C containing G. These semigroups Γ can be viewed as non-commutative analogues of Siegel domains, where G ⊂ Γ plays the role of the skeleton. The aim of this paper is to construct analogues of Hardy spaces H 2 on semigroups Γ and to study their properties.