Let be a continuous map of the closure of the open unit disc of into a unital associative Banach algebra , whose restriction to is holomorphic, and which satisfies the condition whereby for all and whenever (where is the spectrum of any ). One of the basic results of the present paper is that is , that is to say, is then a compact subset of that does not depend on for all . This fact will be applied to holomorphic self-maps of the open unit ball of some -algebra...
In the paper we characterize the two-dimensional generalized Berwald manifolds in terms of the classical setting of Finsler surfaces (Berwald frame, main scalar etc.). As an application we prove that if a Landsberg surface is a generalized Berwald manifold then it must be a Berwald manifold. Especially, we reproduce Wagner's original result in honor of the 75th anniversary of publishing his pioneering work about generalized Berwald manifolds.
We prove that if X is an infinite-dimensional Banach space with smooth partitions of unity then X and X∖ K are diffeomorphic for every weakly compact set K ⊂ X.
Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space of any manifold . The name comes from the fact that various elements of the geometry of are constructed via lifting of the corresponding elements of the geometry of . In this note, we construct a general algebraic framework for lifted geometry which can be applied to various “infinite dimensional spaces” associated to . In order to define a lifted...
In this paper, firstly we study some left invariant Riemannian metrics on para-hypercomplex 4-dimensional Lie groups. In each Lie group, the Levi-Civita connection and sectional curvature have been given explicitly. We also show these spaces have constant negative scalar curvatures. Then by using left invariant Riemannian metrics introduced in the first part, we construct some left invariant Randers metrics of Berwald type. The explicit formulas for computing flag curvature have been obtained in...