Related papers: Controlling almost-invariant halfspaces in both re…

This article is a continuation of a paper of the first author \cite{F} about complex structures on real Banach spaces. We define a notion of even infinite dimensional real Banach space, and prove that there exist even spaces, including HI…

We prove the existence of the invariant subspaces of some operators in a real Banach space. For example, linear isometries have invariant subspaces

We consider real spaces only. Definition. An operator $T:X\to Y$ between Banach spaces $X$ and $Y$ is called a Hahn-Banach operator if for every isometric embedding of the space $X$ into a Banach space $Z$ there exists a norm-preserving…

We study a semigroup $\phi$ of linear operators acting on a Banach space $X$ which satisfies the condition $\codim X_0<\infty$, where $X_0=\{x\in X \mid \phi_t(x)\underset{t\to\infty}\longrightarrow 0\}.$ We show that $X_0$ is closed under…

For a bounded function $f$ from the unit sphere of a closed subspace $X$ of a Banach space $Y$, we study when the closed convex hull of its spatial numerical range $W(f)$ is equal to its intrinsic numerical range $V(f)$. We show that for…

We address a question by Henry Towsner about the possibility of representing linear operators between ultraproducts of Banach spaces by means of ultraproducts of nonlinear maps. We provide a bridge between these "accessible" operators and…

A reflexive hereditarily indecomposable Banach space $\mathfrak{X}_{_{^\text{ISP}}}$ is presented, such that for every $Y$ infinite dimensional closed subspace of $\mathfrak{X}_{_{^\text{ISP}}}$ and every bounded linear operator…

We show that operators on a separable infinite dimensional Banach space $X$ of the form $I +S$, where $S$ is an operator with dense generalised kernel, must lie in the norm closure of the hypercyclic operators on $X$, in fact in the closure…

We show that if $X$ and $Y$ are Banach spaces, where $Y$ is separable and polyhedral, and if $T:X \to Y$ is a bounded linear operator such that $T^*(Y^*)$ contains a boundary $B$ of $X$, then $X$ is separable and isomorphic to a polyhedral…

If $X$ is a separable infinite dimensional Banach space, we construct a bounded and linear operator $R$ on $X$ such that $$ A_R=\{x \in X, \|R^tx\| \rightarrow \infty\} $$ is not dense and has non empty interior with the additional property…

In this paper, we introduce the notions of $\alpha$-quasicomplemented and totally $\alpha$-quasicomplemented subspaces and we established some results under these contexts. We show, for example, that if $X$ is a separable or reflexive…

Let $X$ be a Banach space and $\mathcal A$ be the Banach algebra $B(X)$ of bounded (i.e. continuous) linear transformations (to be called operators) on $X$ to itself. Let $\mathcal E$ be the set of idempotents in $\mathcal A$ and $\mathcal…

Given a separable Banach space $E$, we construct an extremely non-complex Banach space (i.e. a space satisfying that $\|Id + T^2\|=1+\|T^2\|$ for every bounded linear operator $T$ on it) whose dual contains $E^*$ as an $L$-summand. We also…

The objective of this article is to study nearly invariant subspaces of the backward shift operator on the real Hardy space. We also investigate nearly invariant subspaces with finite defect, and as a consequence, provide a characterization…

Given a Banach space $E$, we ask which closed subspaces may be realised as the kernel of a bounded operator $E \rightarrow E$. We prove some positive results which imply in particular that when $E$ is separable every closed subspace is a…

A banach space X is a normed vector space, which is complete with respect to the metric induced by the norm. Given a bounded linear operator T acting on a banach space X, T is said to attain its norm if there is a unit vector z in X, such…

We review recent work connected with the invariant subspace problem for operators, in particular new developments in the last 15 years. In particular, we include discussions of almost-invariant subspaces, universal operators, specific…

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic…

Let X and Y be two infinite-dimensional Banach spaces. If X is crudely finitely representable in every finite-codimensional subspace of Y, then any proper subset of X almost bi-Lipschitz embeds into Y, in a sense quite close to that of F.…

For a power bounded or polynomially bounded operator $T$ sufficient conditions for the existence of a nontrivial hyperinvariant subspace are given. The obtained hyperinvariant subspaces of $T$ have the form of the closure of the range of…