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We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators $\Gamma$ on a Hilbert space. Our assumption on $\Gamma$ is expressed in terms of…
The purpose of this paper is to study stochastic evolution inclusions of the form \begin{align*} \eta(t,z) N_{\Theta}(dt \otimes z)\in dX(t)+\mathcal{A} X(t)dt, \end{align*} where $\mathcal{A}$ is a multi-valued operator acting on a…
Let $\mathscr{B}(X)$ denote the Banach algebra of bounded operators on $X$, where~$X$ is either Tsirelson's Banach space or the Schreier space of order $n$ for some $n\in\mathbb N$. We show that the lattice of closed ideals…
In this paper, we introduce a new class of subsets of bounded linear operators between Banach spaces which is p-version of the uniformly completely continuous sets. Then, we study the relationship between these sets with the equicompact…
In a previous paper the second author introduced a compact topology on the space of closed ideals of a unital Banach algebra A. If A is separable then this topology is either metrizable or else neither Hausdorff nor first countable. Here it…
We consider the action of finitely truncated singular integral operators on functions taking values in a Banach space. Such operators are bounded for any Banach space, but we show a quantitative improvement over the trivial bound in any…
This thesis presents a formalization of martingales in arbitrary Banach spaces using Isabelle/HOL. We begin by examining formalizations in prominent proof repositories and extend the definition of the conditional expectation operator from…
We consider linear bounded operators acting in Banach spaces with a basis, such operators can be represented by an infinite matrix. We prove that for an invertible operator there exists a sequence of invertible finite-dimensional operators…
We prove when a Banach ideal of linear operators defined, or characterized, by the transformation of vector-valued sequences is maximal. Known results are recovered as particular cases and new information is obtained. To accomplish this…
Given a category of objects, it is both useful and important to know if all the objects in the category may be realised as sub-objects -- via morphisms in the given category -- of a single object in that category enjoying some nice…
We consider a finite or countable collection of one-dimensional Brownian particles whose dynamics at any point in time is determined by their rank in the entire particle system. Using Transportation Cost Inequalities for stochastic…
Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence (…
We study $M$-ideals of compact operators by means of the property~$(M)$ introduced in \cite{Kal-M}. Our main result states for a separable Banach space $X$ that the space of compact operators on $X$ is an $M$-ideal in the space of bounded…
Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. We treat questions of characterizing which of the spaces from a given…
Let $X$ be a Banach space with a basis $(e_k)_k$ and biorthogonals $(e^\ast_k)_k$. An operator on $X$ is said to have a $\textit {large diagonal}$ if $\inf\limits_{k} |e_k^\ast(T(e_k))| > 0$. The basis $(e_k)_k$ is said to have the $\textit…
In this work we will consider integral equations defined on the whole real line and look for solutions which satisfy some certain kind of asymptotic behavior. To do that, we will define a suitable Banach space which, to the best of our…
In this paper, we study the differentiation operator acting on discrete function spaces; that is spaces of functions defined on an infinite rooted tree. We discuss, through its connection with composition operators, the boundedness and…
We consider the optimal stopping problem for a Gauss-Markov process conditioned to adopt a prescribed terminal distribution. By applying a time-space transformation, we show it is equivalent to stopping a Brownian bridge pinned at a random…
For each sequence X of finite-dimensional Banach spaces there exists a sequence H of finite connected nweighted graphs with maximum degree 3 such that the following conditions on a Banach space Y are equivalent: (1) Y admits uniformly…
This work provides a novel convergence analysis for stochastic optimization in terms of stopping times, addressing the practical reality that algorithms are often terminated adaptively based on observed progress. Unlike prior approaches,…