Related papers: Extension operators and nonlinear structure of Ban…

In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In…

We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined…

We construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and $C^1$ functions. This way we prove more directly a result by Lee and Naor and we generalize the $C^1$…

The Roper--Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension…

The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then…

We provide an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The…

Within Bishop-style constructive mathematics we study the classical McShane-Whitney theorem on the extendability of real-valued Lipschitz functions defined on a subset of a metric space. Using a formulation similar to the formulation of…

While the theory of matrix-weighted function spaces is well established, the majority of previous results in the infinite-dimensional operator-valued setting deal with "no go" theorems, showing the impossibility of some prospective…

In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this is that on such a Banach space,…

We present an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The…

Let us consider a Banach space $X$ with the property that every real-valued Lipschitz function $f$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with $\Lip(g)\le C \Lip(f)$ (with $C$ depending only on the space…

Valadier and Hensgen proved independently that the restriction of functional $\phi(x)=\int_{0}^{1}x(t)dt,\,\,x\in L^{\infty}([0,1])$ on the space of continuous functions $C([0,1])$ admits a singular extension back to the whole space…

We introduce a modified version of the Whitney extension operators for collections of functions from a closed subset of $\mathbb{R}^n$ into scales of Banach spaces with smoothing operators. We prove an extension theorem for collections…

Let $X$ be a normed space of a finite dimension at least two, and $C\subsetneq X$ a closed convex set with nonempty interior. We are interested in extending Lipschitz quasiconvex functions on $C$ to quasiconvex functions on $X$. We show…

Let $E, F, E_0$ be Banach spaces, with $E_0$ a subspace of $E$. For a maximal Banach operator ideal $\mathcal{A}$, we show that a linear operator from $E_0$ to $F$ can be extended to a linear operator from $E$ to $F$ that belongs to…

We study the isometric extension problem for H\"{o}lder maps from subsets of any Banach space into $c_0$ or into a space of continuous functions. For a Banach space $X$, we prove that any $\alpha$-H\"{o}lder map, with $0<\alpha\leq 1$, from…

A multiple-valued function $f:X\to {\bf Q}_Q(Y)$ is essentially a rule assigning $Q$ unordered and non necessarily distinct elements of $Y$ to each element of $X$. We study the Lipschitz extension problem in this context by using two…

We consider subsets $S$ of a metric space $M$ such that Lipschitz mappings defined on $S$ can be extended to Lipschitz mappings on $M$, and we show that the union of such subsets has the same property under appropriate geometric conditions.…

The Lipschitz extension modulus $e(M)$ of a metric space $M$ is the infimum over $L\ge 1$ such that for any Banach space $Z$ and any $C\subset M$, any 1-Lipschitz function $f:C\to Z$ can be extended to an $L$-Lipschitz function $F:M\to Z$.…

We study a new bi-Lipschitz invariant \lambda(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are enlarged by a factor…