Related papers: Interpolation of compact Lipschitz operators

For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$, ${\mathcal K}\circ{\mathcal A}$ and ${\mathcal K}\circ{\mathcal A}\circ{\mathcal K}$, where $\mathcal K$ is the ideal of compact…

We characterize when an Orlicz space $L^A$ is almost compactly (uniformly absolutely continuously) embedded into a Lorentz space $L^{p,q}$ in terms of a balance condition involving parameters $p,q\in[1,\infty]$, and a Young function $A$. In…

Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with an increasing filtration $(\mathcal{M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal{M}$. For $1\leq p \leq\infty$, let $\mathcal{H}_p^c(\mathcal{M})$…

We consider K-interpolation methods involving slowly varying functions. Let $\overline{A}_{\theta,*}^{\mathcal{L}}$ and $\overline{A}_{\theta,*}^{\mathcal{R}}$ $(0\leq\theta\leq1)$ be the so called ${\mathcal{L}}$ or ${\mathcal{R}}$…

Suppose that the linear operator $T$ maps $X_0$ compactly to $Y_0$ and also maps $X_1$ boundedly to $Y_1$. We deal once again with the 51 year old question of whether $T$ also always maps the complex interpolation space $[X_0,X_1]_\theta$…

We give a sufficient condition for a pair of Banach spaces $(X,Y)$ to have the following property: whenever $W_1 \subseteq X$ and $W_2 \subseteq Y$ are sets such that $\{x\otimes y: \, x\in W_1, \, y\in W_2\}$ is weakly precompact in the…

Let $\mathcal{P}$ be a class of Banach spaces and let $T=\{T_\alpha\}_{\alpha\in A}$ be a set of metric spaces. We say that $T$ is a set of {\it test-spaces} for $\mathcal{P}$ if the following two conditions are equivalent: (1)…

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat{f} : \mathcal F(M) \to \mathcal F(N)$ between their corresponding Lipschitz-free spaces. In this paper,…

Given an inner function $\theta$ on the unit disk, let $K^p_\theta:=H^p\cap\theta\bar z\bar{H^p}$ be the associated star-invariant subspace of the Hardy space $H^p$. Also, we put $K_{*\theta}:=K^2_\theta\cap{\rm BMO}$. Assuming that…

For pointed compact metric spaces $(X,d)$, we address the biduality problem as to when the space of Lipschitz functions $\mathrm{Lip}_0(X,d)$ is isometrically isomorphic to the bidual of the space of little Lipschitz functions…

Given $E_0, E_1, E, F$ rearrangement invariant spaces, $a, b, b_0, b_1$ slowly varying functions and $0\leq \theta_0<\theta_1\leq 1$, we characterize the interpolation space $$(\overline{X}_{\theta_0,b_0,E_0}, \overline{X}^{\mathcal…

We prove that the Lipschitz-free space over a Banach space $X$ of density $\kappa$, denoted by $\mathcal{F}(X)$, is linearly isomorphic to its $\ell_1$-sum $\left(\bigoplus_{\kappa}\mathcal{F}(X)\right)_{\ell_1}$. This provides an extension…

The main aim of this paper is to develop a general approach, which allows to extend the basics of Brudnyi-Kruglyak interpolation theory to the realm of quasi-Banach lattices. We prove that all $K$-monotone quasi-Banach lattices with respect…

The Banach space $\mathcal{P}({}^2X)$ of $2$-homogeneous polynomials on the Banach space $X$ can be naturally embedded in the Banach space ${{\rm Lip}_0}(B_X)$ of real-valued Lipschitz functions on $B_X$ that vanish at $0$. We investigate…

We improve the known results about the complexity of the relation of isomorphism between separable Banach spaces up to Borel reducibility, and we achieve this using the classical spaces $c_0$, $\ell_p$ and $L_p$, $1 \leq p <2$. More…

We consider the problem of complex interpolation of certain Hardy-type subspaces of K\"othe function spaces. For example, suppose $X_0$ and $X_1$ are K\"othe function spaces on the unit circle $\bold T,$ and let $H_{X_0}$ and $H_{X_1}$ be…

We consider operators T : M_0 -> Z and T : M -> Z, where Z is a Banach space and (M_0, M) is a pair of Banach spaces belonging to a general construction in which M is defined by a "big-O" condition and M_0 is given by the corresponding…

A Banach space operator $A\in B({\cal{X}})$ is polaroid, $A\in {\cal{P}}$, if the isolated points of the spectrum $\sigma(A)$ are poles of the operator; $A$ is hereditarily polaroid, $A\in{\cal{HP}}$, if every restriction of $A$ to a closed…

Let $B_p(s)$ be an analytic Besov type space. Let $M(B_p(s))$ be the class of multipliers of $B_p(s)$ and let $F(p, p-2, s)$ be the M\"obius invariant subspace generated by $B_p(s)$. In this paper, when $0<s<1$ and $\max\{s, 1-s\}<p\leq 1$,…

We give sufficient conditions for a metric space to bilipschitz embed in L_1. In particular, if X is a length space and there is a Lipschitz map u:X--->R such that for every interval I in R, the connected components of the inverse image…