Orbit structure of non-compact hermitian symmetric spaces

SiaX ₀ uno spazio simmetrico hermitiano di tipo non compatto,x ₀ un punto diX ₀ eK ₀ il sottogruppo d'isotropia dix ₀. In questo articolo viene studiata approfonditamente la struttura dello spazio delle orbite dell'azione diK ₀ suX ₀. Ad ogni puntox∈X ₀ vengono associatir=rankX ₀ numeri reali non negative dettimoduli, che parametrizzano lo spazio delle orbite. Usando questi moduli, è possibile stabilire effettivamente quando due orbite sono isomorfe e, più in generale, quando esiste una applicazione fra due orbite che commuta con l'azione diK ₀. Inoltre è anche possibile determinare la dimensione di ogni orbita, e descrivere la struttura del bordo di ?ilov diX ₀, generalizzando alcuni risultati di Wolf, Korányi e Harris. Infine, viene data la realizzazione concreta delle definizioni introdotte e dei teoremi dimostrati nel caso dei quattro domini classici nella realizzazione di Cartan.     

Numerical peak holomorphic functions on Banach spaces

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let Ab(BX:X) be the Banach space of all bounded continuous functions f on the unit ball BX of a Banach space X and their restrictions to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense Gδ-subset of Ab(BX:X). We also prove that if X is a smooth Banach space with the Radon-Nikodým property, then the set of all numerical strong peak functions is dense in Ab(BX:X). In particular, when X=Lp(μ)(1<p<∞) or X=?₁, it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense Gδ-subset of Ab(BX:X). As an application, the existence and properties of numerical boundary of Ab(BX:X) are studied. Finally, the numerical peak function in Ab(BX:X) is characterized when X=C(K) and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.     

Metrizability of spaces of holomorphic functions

In this paper we prove that if U is an open subset of a metrizable locally convex space E of infinite dimension, the space H(U) of all holomorphic functions on U, endowed with the Nachbin-Coeuré topology τδ, is not metrizable. Our result can be applied to get that, for all usual topologies, H(U) is metrizable if and only if E has finite dimension.     

Vector spaces of non-extendable holomorphic functions

In this paper, we analyze the linear structure of the family H _e (G) of holomorphic functions on a domain G of the complex plane that are not analytically continuable beyond the boundary of G. We prove that H _e (G) contains, except for zero, a dense algebra; and, under appropriate conditions, the subfamily of H _e (G) consisting of boundary-regular functions contains dense vector spaces with maximal dimension as well as infinite dimensional closed vector spaces and large algebras. We also consider the case in which G is a domain of existence in a complex Banach space. The results obtained complete or extend a number of previous results by several authors.     

Homogeneous operators on Hilbert spaces of holomorphic functions

In this paper we construct a large class of multiplication operators on reproducing kernel Hilbert spaces which are homogeneous with respect to the action of the Möbius group consisting of bi-holomorphic automorphisms of the unit disc D. Indeed, this class consists of exactly those operators for which the associated unitary representation of the universal covering group of the Möbius group is multiplicity free. For every m∈N we have a family of operators depending on m+1 positive real parameters. The kernel function is calculated explicitly. It is proved that each of these operators is bounded, lies in the Cowen-Douglas class of D and is irreducible. These operators are shown to be mutually pairwise unitarily inequivalent.     

Hilbert spaces of holomorphic functions on bounded domains

In [6] the notion of Hilbert-Stein complex spaces was introduced. The purpose of this note is to illustrate the conjecture that any Stein space whose points can be separated by bounded holomorphic functions, is Hilbert-Stein. We prove this fact for the following kinds of bounded domains of holomorphyD in C ⁿ : 1)n=1 and ₁(D) is finitely generated, 2) the group of automorphisms ofD in the compactopen topology is compact, 3)D is strongly pseudoconvex with smooth boundary andn2, 4)D is symmetric of type I.     

Integral representation of holomorphic functions on Banach spaces

In this paper we discuss the problem of integral representation of analytic functions over a complex Banach space E. We obtain, for a wide class of functions, integral representations of the form

     

Fredholm composition operators on spaces of holomorphic functions

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorC on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC ² boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if is a holomorphic automorphism.     

Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces

Dedicated to professor K. Oikawa on his 60th birthday