We study Kalton's theorem on the unconditional convergence of series of compact operators and we use some matrix techniques to obtain sufficient conditions, weaker than the previous one, on the convergence and unconditional convergence of series of compact operators.
In this note we investigate the relationship between the convergence of the sequence of sums of independent random elements of the form (where takes the values with the same probability and belongs to a real Banach space for each ) and the existence of certain weakly unconditionally Cauchy subseries of .
We prove that the converse of Theorem 9 in "On generalized inverses in C*-algebras" by Harte and Mbekhta (Studia Math. 103 (1992)) is indeed true.
For large classes of complex Banach spaces (mainly operator spaces) we consider orbits of finite rank elements under the group of linear isometries. These are (in general) real-analytic submanifolds of infinite dimension but of finite CR-codimension. We compute the polynomial convex hull of such orbits explicitly and show as main result that every continuous CR-function on has a unique extension to the polynomial convex hull which is holomorphic in a certain sense. This generalizes to infinite...
We consider scalar products on a given Hilbert space parametrized by bounded positive and invertible operators defined on this space, and orthogonal projectors onto a fixed closed subspace of the initial Hilbert space corresponding to these scalar products. We show that the projector is an analytic function of the scalar product, we give the explicit formula for its Taylor expansion, and we prove some algebraic formulas for projectors.
For a (DF)-space E and a tensor norm α we investigate the derivatives of the tensor product functor from the category of Fréchet spaces to the category of linear spaces. Necessary and sufficient conditions for the vanishing of , which is strongly related to the exactness of tensored sequences, are presented and characterizations in the nuclear and (co-)echelon cases are given.