2000 Mathematics Subject Classification: 46B50, 46B70, 46G12.A new measure of noncompactness on Banach spaces is defined from the Hausdorff measure of noncompactness, giving a quantitative version of a classical result by R. S. Phillips. From the main result, classical results are obtained now as corollaries and we have an application to interpolation theory of Banach spaces.
The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function is absolutely continuous. As a consequence we obtain that the McShane product integral of over exists and is invertible if and only if is Bochner integrable...
We study spaces of analytic functions generated by homogeneous polynomials from the dual space to the symmetric Hilbertian tensor product of a Hilbert space. In particular, we introduce an analogue of the classical Hardy space H² on the Hilbert unit ball and investigate spectral decomposition of unitary operators on this space. Also we prove a Wiener-type theorem for an algebra of analytic functions on the Hilbert unit ball.
A Hille-Yosida Theorem is proved on convenient vector spaces, a class, which contains all sequentially complete locally convex spaces. The approach is governed by convenient analysis and the credo that many reasonable questions concerning strongly continuous semigroups can be proved on the subspace of smooth vectors. Examples from literature are reconsidered by these simpler methods and some applications to the theory of infinite dimensional heat equations are given.
A class of Banach spaces of compact operators in Hilbert spaces is introduced, and the holomorphic automorphism groups of the unit balls of these spaces are investigated.
We endow the direct sum of two complex Banach spaces with a suitable norm, and we investigate the orbit of the origin for the group of holomorphic automorphisms of the outcoming unit ball.
We introduce a decomposition of holomorphic functions on Fréchet spaces which reduces to the Taylor series expansion in the case of Banach spaces and to the monomial expansion in the case of Fréchet nuclear spaces with basis. We apply this decomposition to obtain examples of Fréchet spaces E for which the τ_{ω} and τ_{δ} topologies on H(E) coincide. Our result includes, with simplified proofs, the main known results-Banach spaces with an unconditional basis and Fréchet nuclear spaces with DN [2,...
In this article we show that a number of apparently different properties coincide on the set of holomorphic functions on a strict inductive limit (all inductive limits are assumed to be countable and proper) of Banach spaces and that they are all satisfied only in the trivial case of a strict inductive limit of finite dimensional spaces. Thus the linear properties of a strict inductive limit of Banach spaces rarely translate themselves into holomorphic properties.
The holomorphic isometries for the Kobayashi metric of Cartan domains of type four are characterized.
A previous paper was devoted to the construction of non-trivial holomorphic families of holomorphic isometries for the Carathéodory metric of a bounded domain in a complex Banach space, fixing a point in the domain. The present article shows that such a family cannot exist if it contains a strongly continuous one parameter semigroup.