We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.

Let $X$ be a completely regular Hausdorff space, $E$ a real Banach space, and let $C_b(X,E)$ be the space of all $E$-valued bounded continuous functions on $X$. We study linear operators from $C_b(X,E)$ endowed with the strict topologies ${\beta }_z$$(z=\sigma ,...$

The paper introduces a notion of strictly convex metric space and strictly convex metric space with round balls. These objects generalize the well known concept of strictly convex Banach space. We prove some fixed point theorems in strictly convex metric spaces with round balls.

Let ${\left\{\beta \left(n\right)\right\}}_{n=0}^{\infty }$ be a sequence of positive numbers and $1\le p<\infty$. We consider the space $H^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\infty }\widehat{f}\left(n\right)z^n$ such that ${\sum }_{n=0}^{\infty }{\left|\widehat{f}\left(n\right)\right|}^p\beta {\left(n\right)}^p<\infty$. We investigate strict cyclicity of $H_p^{\infty }\left(\beta \right)$, the weakly closed algebra generated by the operator of multiplication by $z$ acting on $H^p\left(\beta \right)$, and determine the maximal ideal space, the dual space and the reflexivity of the algebra $H_p^{\infty }\left(\beta \right)$. We also give a necessary condition for a composition operator to be bounded on $H^p\left(\beta \right)$ when $H_p^{\infty }\left(\beta \right)$ is strictly cyclic.

Properties of strictly singular operators have recently become of topical interest because the work of Gowers and Maurey in [GM1] and [GM2] gives (among many other brilliant and surprising results, such as those in [G1] and [G2]) Banach spaces on which every continuous operator is of form λ I + S, where S is strictly singular. So if strictly singular operators had invariant subspaces, such spaces would have the property that all operators on them had invariant subspaces. However, in this paper we...

An integral Markov operator $P$ appearing in biomathematics is investigated. This operator acts on the space of probabilistic Borel measures. Let $\mu$ and $\nu$ be probabilistic Borel measures. Sufficient conditions for weak and strong convergence of the sequence $(P^n\mu -P^n\nu )$ to $0$ are given.

Let J be an abelian topological semigroup and C a subset of a Banach space X. Let L(X) be the space of bounded linear operators on X and Lip(C) the space of Lipschitz functions ⨍: C → C. We exhibit a large class of semigroups J for which every weakly continuous semigroup homomorphism T: J → L(X) is necessarily strongly continuous. Similar results are obtained for weakly continuous homomorphisms T: J → Lip(C) and for strongly measurable homomorphisms T: J → L(X).