Let $E=\bigcup _{n=1}^{\infty }I_{n}$ be the union of infinitely many disjoint closed intervals where $I_{n}=[a_{n}$, $b_{n}]$, $0<a_{1}<b_{1}<a_{2}<b_{2}<\dots <b_{n}<\dots $, $\lim _{n\rightarrow \infty }b_{n}=\infty .$ Let $\alpha (t)$ be a nonnegative function and $\{\lambda _{n}\}_{n=1}^{\infty }$ a sequence of distinct complex numbers. In this paper, a theorem on the completeness of the system $\{t^{\lambda _{n}}\log ^{m_{n}}t\}$ in $C_{0}(E)$ is obtained where $C_{0}(E)$ is the weighted Banach space consists of complex functions continuous on $E$ with $f(t){\rm e}^{-\alpha (t)}$ vanishing at infinity.
Let $T$ be an operator acting on a Banach space $X$. We show that between extensions of $T$ to some Banach space $Y\supset X$ which do not increase the defect spectrum (or the spectrum) it is possible to find an extension with the minimal possible defect spectrum.
Let X be a normed linear space. We investigate properties of vector functions F : [a, b] → X of bounded convexity. In particular, we prove that such functions coincide with the delta-convex mappings admitting a Lipschitz control function, and that convexity Kb a F is equal to the variation of F ′ + on [a, b). As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.
The abstract Perron-Stieltjes integral in the Kurzweil-Henstock sense given via integral sums is used for defining convolutions of Banach space valued functions. Basic facts concerning integration are preseted, the properties of Stieltjes convolutions are studied and applied to obtain resolvents for renewal type Stieltjes convolution equations.
In this paper we analyse a definition of a product of Banach spaces that is naturally associated by duality with a space of operators that can be considered as a generalization of the notion of space of multiplication operators. This dual relation allows to understand several constructions coming from different fields of functional analysis that can be seen as instances of the abstract one when a particular product is considered. Some relevant examples and applications are shown, regarding pointwise products of Banach function spaces, spaces of integrable functions with respect to vector measures, spaces of operators, multipliers on Banach spaces of analytic functions and spaces of Lipschitz functions., Enrique A. Sánchez Pérez., and Obsahuje seznam literatury
We establish the Doob inequality for martingale difference arrays and provide a sufficient condition so that the strong law of large numbers would hold for an arbitrary array of random elements without imposing any geometric condition on the Banach space. Some corollaries are derived from the main results, they are more general than some well-known ones.
Let $X$ be a Banach space with the Grothendieck property, $Y$ a reflexive Banach space, and let $X\check{\otimes}_{\varepsilon} Y$ be the injective tensor product of $X$ and $Y$. \item {(a)} If either $X^{\ast \ast }$ or $Y$ has the approximation property and each continuous linear operator from $X^\ast $ to $Y$ is compact, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property. \item {(b)} In addition, if $Y$ has an unconditional finite dimensional decomposition, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property if and only if each continuous linear operator from $X^\ast $ to $Y$ is compact.