Related papers: Extension operators and nonlinear structure of Ban…

We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we…

Let $\{\tau_j\}_{j=1}^n$ be a collection in a finite dimensional Banach space $\mathcal{X}$ of dimension $d$ and $\{f_j\}_{j=1}^n$ be a collection in $\mathcal{X}^*$ (dual of $\mathcal{X}$) such that $f_j(\tau_j) =1$, $\forall 1\leq j\leq…

Let us consider a Riemannian manifold $M$ (either separable or non-separable). We prove that, for every $\epsilon>0$, every Lipschitz function $f:M\rightarrow\mathbb R$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$…

Let $S$ be a compact Hausdorff space and $X$ a complex manifold. We consider the space $C(S,X)$ of continuous maps $S\to X$, and prove that any bounded holomorphic function on this space can be continued to a holomorphic function, possibly…

We consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by…

We show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing…

It is shown that there exist Banach spaces $X,Y$, a $1$-net $\mathscr{N}$ of $X$ and a Lipschitz function $f:\mathscr{N}\to Y$ such that every $F:X\to Y$ that extends $f$ is not uniformly continuous.

The main result of this paper is a proof of the continuity of a family of integral functionals defined on the space of functions of bounded variation with respect to a topology under which smooth functions are dense. These functionals occur…

Let $f:X\to X$ be an invertible Lipschitz transformation on a compact metric space $X$. Given a H\"{o}lder continuous invertible operator cocycles on a Banach space and an $f$-invariant ergodic measure, this paper establishes the H\"{o}lder…

Let $A$ be a Banach algebra and $I$ a dense ideal in $A$. A natural question in the theory of operator algebras is whether the property that all derivations $D: A \to I$ are inner (implemented by elements in $I$) implies that all…

Let ${T_1,...,T_l}$ be a collection of differential operators with constant coefficients on the torus $\mathbb{T}^n$. Consider the Banach space $X$ of functions $f$ on the torus for which all functions $T_j f$, $j=1,...,l$, are continuous.…

The paper deals with extension of bounded bilinear maps$.$ It gives a necessary and sufficient condition for extending a bounded bilinear map on the Cartesian product of subspaces of Banach spaces$.$ This leads to a full characterization…

We investigate the problem of existence of a bounded extension to $C(K)$ of a bounded $c_0(I)$-valued operator $T$ defined on the subalgebra of $C(K)$ induced by a continuous increasing surjection $\phi:K\to L$, where $K$ and $L$ are…

Under certain hypotheses on the Banach space $X$, we prove that the set of analytic functions in $\mathcal{A}_u(X)$ (the algebra of all holomorphic and uniformly continuous functions in the ball of $X$) whose Aron-Berner extensions attain…

This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional…

Let $E_1, \ldots, E_m$ be (non necessarily Archimedean) Riesz spaces, let $F$ be an Archimedean Riesz space and let $A \colon E_1 \times \cdots \times E_m \to F$ be a regular disjointness preserving $m$-linear operator. We prove that all…

We provide sufficient conditions on a Banach space $X$ in order that there exist norm attaining operators of rank at least two from $X$ into any Banach space of dimension at least two. For example, a rather weak such condition is the…

In the literature surrounding the theory of Banach spaces, considerable effort has been invested in exploring the conditions on a Banach space X that characterise X as being an inner product space or as a linearly isomorphic copy of a…

Our note is a complement to recent articles \cite{JS1} (2011) and \cite{JS2} (2013) by M. Jim\'enez-Sevilla and L. S\'anchez-Gonz\'alez which generalise (the basic statement of) the classical Whitney extension theorem for $C^1$-smooth real…

Notion of frames and Bessel sequences for metric spaces have been introduced. This notion is related with the notion of Lipschitz free Banach spaces. \ It is proved that every separable metric space admits a metric $\mathcal{M}_d$-frame.…