When U is the open unit ball of a separable Banach space E, we show that , the predual of the space of bounded holomorphic mappings on U, has the bounded approximation property if and only if E has the bounded approximation property.
I will explain basic concepts/problems of complex analysis in infinite dimensions, and survey the few approaches that are available to solve those problems.
We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ||Id + P|| = 1 + ||P|| is satisfied for all weakly compact polynomials P: X → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation for polynomials P: X → X. We show that this equation holds for every polynomial on the complex space X =...
We obtain conditions on a JB*-algebra X so that the canonical embedding of X into its associated quasi-invertible manifold has dense range. We prove that if a JB* has this density property then the quasi-invertible manifold is homogeneous for biholomorphic mappings. Explicit formulae for the biholomorphic mappings are also given.
The aim of this paper is to start a systematic investigation of the existence of angular limits and angular derivatives of holomorphic maps of infinite dimensional Siegel domains in -algebras. Since -algebras are natural generalizations of -algebras, -algebras, -algebras, ternary algebras and complex Hilbert spaces, various significant results follow. Examples are given.
We give a simple proof of the result that if D is a (not necessarily bounded) hyperbolic convex domain in then the set V of fixed points of a holomorphic map f:D → D is a connected complex submanifold of D; if V is not empty, V is a holomorphic retract of D. Moreover, we extend these results to the case of convex domains in a locally convex Hausdorff vector space.
It is proved that the Levi problem for certain locally convex Hausdorff spaces over with a finite dimensional Schauder decomposition (for example for Fréchet or Silva spaces with a Schauder basis) the Levi problem has a solution, i.e. every pseudoconvex domain spread over is a domain of existence of an analytic function. It is also shown that a pseudoconvex domain spread over a Fréchet space or a Silva space with a finite dimensional Schauder decomposition is holomorphically convex and satisfies...
Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals...
The multiplier for the weak McShane integral which has been introduced by M. Saadoune and R. Sayyad (2014) is characterized.