Multiple brake orbits in bounded convex symmetric domains

In this paper, we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in .     

Boundary structure of bounded symmetric domains

Let D be a bounded symmetric domain realized as the open unit ball of a complex Banach space E and denote for every by s _a the symmetry of D about a. We show that for every boundary point the locally uniform limit s _c := lim _{c → c} s _a exists as holomorphic map s _c :D→E and has values in the boundary of D. Received: 7 August 1999 / Revised version: 3 November 1999     

Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains

We prove that any proper holomorphic mapping from to is necessarily a totally geodesic isometric embedding with respect to their Bergman metrics and therefore is the standard linear embedding up to their automorphisms. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics. Received: 31 October 2000; in final form: 2 July 2001/ Published online: 4 April 2002     

Complex geodesics in a class of convex bounded Reinhardt domains

We find explicitly all complex geodesics in a class of convex bounded Reinhardt domains ofC ⁿ, which are a generalization of complex ellipsoids. Entrata in Redazione il 19 maggio 1998.     

Analytic functions of bounded mean oscillation and the Bloch space

In this paper we show that the closure of the space BMOA of analytic functions of bounded mean oscillation in the Bloch spaceB is the image P(U) of space of all continuous functions on the maximal ideal space ofH under the Bergman projection P. It is proved that the radial growth of functions in P(U) is slower than the iterated logarithm studied by Makarov. So some geometric conditions are given for functions in P(U), which we can easily use to construct many Bloch functions not in P(U).     

Compact Toeplitz operators via the Berezin transform on bounded symmetric domains

Let be an irreducible bounded symmetric domain of genusp, h(x, y) its Jordan triple determinant, andA ² () the standard weighted Bergman space of holomorphic functions on square-integrable with respect to the measureh(z, z) ^–p dz. Extending the recent result of Axler and Zheng for =D, =p=2 (the unweighted Bergman space on the unit disc), we show that ifS is a finite sum of finite products of Toeplitz operators onA ² () and is sufficiently large, thenS is compact if and only if the Berezin transform ofS tends to zero asz approaches . An analogous assertion for the Fock space is also obtained.The author's research was supported by GA AV R grant A1019701 and GA R grant 201/96/0411.     

Bloch constants of bounded symmetric domains

Let and be two irreducible bounded symmetric domains in the complex spaces and respectively. Let be the Euclidean metric on and the Bergman metric on . The Bloch constant is defined to be the supremum of , taken over all the holomorphic functions and , and nonzero vectors . We find the constants for all the irreducible bounded symmetric domains and . As a special case we answer an open question of Cohen and Colonna.

     

Asymptotic expansions and Hua-harmonic functions on bounded homogeneous domains

Let D be a homogeneous Siegel domain of type II. We prove that every bounded Hua-harmonic function F on D is pluriharmonic. The proof is based on asymptotic expansion of F. The author was partly supported by KBN grant 1 P03A 018 26, Foundation for Polish Sciences, Subsidy 3/99, and by the European Commission via the TMR Network ``Harmonic Analysis', contract no. ERB FMRX-CT97-0159.     

Dimensions of very ample pluricanonical subsystems for compact quotients of bounded symmetric domains

A compact complex manifold X obtained by taking quotient of a bounded symmetric domain has an ample canonical line bundle. We prove that the dimension of very ample pluricanonical subsystem is strictly bigger than 2n, where n is the dimension of X. Received: 23 June 2000 / Revised version: 30 March 2001     

Gleason's problem for harmonic Bergman and Bloch functions on half-spaces

On the setting of the half-spaceR ^n–1×R ₊, we investigate Gleason's problem for harmonic Bergman and Bloch functions. We prove that Gleason's problem for the harmonicL ^p -Bergman space is solvable if and only ifp>n. We also prove that Gleason's problem for the harmonic (little) Bloch space is solvable.     

Differential systems of type (1,1) on Hermitian symmetric spaces and their solutions

This paper concerns G-invariant systems of second-order differential operators on irreducible Hermitian symmetric spaces G/K. The systems of type (1,1) are obtained from K-invariant subspaces of . We show that all such systems can be derived from a decomposition . Here gives the Laplace-Beltrami operator and is the celebrated Hua system, which has been extensively studied elsewhere. Our main result asserts that for G/K of rank at least two, a bounded real-valued function is annihilated by the system if and only if it is the real part of a holomorphic function. In view of previous work, one obtains a complete characterization of the bounded functions that are solutions for any system of type (1,1) which contains the Laplace-Beltrami operator.     

Bloch spaces on bounded symmetric domains in complex Banach spaces

We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Maximum boundary regularity of bounded Hua-harmonic functions on tube domains

In this article we prove that bounded Hua-harmonic functions on tube domains that satisfy some boundary regularity condition are necessarily pluriharmonic. In doing so, we show that a similar theorem is true on one-dimensional extensions of the Heisenberg group or equivalently on the Siegel upper half-plane.     

Harmonic maps of bounded symmetric domains

Research supported in part by NNSFC, SFECC and ICTP     

A smooth compactification of the space of transfer functions with fixed McMillan degree

It is a classical result of Clark that the space of all proper or strictly properp ×m transfer functions of a fixed McMillan degreed has, in a natural way, the structure of a noncompact, smooth manifold. There is a natural embedding of this space into the set of allp × (m+p) autoregressive systems of degree at mostd. Extending the topology in a natural way we will show that this enlarged topological space is compact. Finally we describe a homogenization process which produces a smooth compactification.This author was supported in part by NSF grant DMS-9201263.     

Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains

The classical integral representation formulas for holomorphic functions defined on pseudoconvex domains in Stein manifolds play an important role in the constructive theory of functions of several complex variables. In this paper, we will show how to construct similar formulas for certain classes of holomorphic functions defined on coverings of such domains.     

Graph homomorphisms and nodal domains

In this paper, we derive some necessary spectral conditions for the existence of graph homomorphisms in which we also consider some parameters related to the corresponding eigenspaces such as nodal domains. In this approach, we consider the combinatorial Laplacian and co-Laplacian as well as the adjacency matrix. Also, we present some applications in graph decompositions where we prove a general version of Fisher’s inequality for G-designs.