Related papers: Counterexamples for interpolation of compact Lipsc…

Motivated by a question of Vincent Lafforgue, we study the Banach spaces $X$ satisfying the following property: there is a function $\vp\to \Delta_X(\vp)$ tending to zero with $\vp>0$ such that every operator $T\colon L_2\to L_2$ with…

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…

To any pair ( M , theta ) where M is a family of finite subsets of N compact in the pointwise topology, and 0<theta < 1 , we associate a Tsirelson-type Banach space T_M^theta . It is shown that if the Cantor-Bendixson index of M is greater…

We characterise the Banach spaces $X$ which are $L_1$-predual as those for which every Lipschitz compact mapping $f:N\longrightarrow X$ admits, for every $\varepsilon>0$ and every $M$ containing $N$, a Lipschitz (compact) extension…

A complex number $\lambda$ is called an extended eigenvalue of a bounded linear operator $T$ on a Banach space $\B$ if there exists a non-zero bounded linear operator $X$ acting on $\B$ such that $XT=\lambda TX$. We show that there are…

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…

This paper extends topics in linear algebra and operator theory for linear transformations on complex vector spaces to those on bicomplex Hilbert and Banach spaces. For example, Definition 3 for the first time defines a bicomplex vector…

For linear operators $L, T$ and nonlinear maps $P$, we describe classes of simple maps $F = I - P T$, $F = L - P$ between Banach and Hilbert spaces, for which no point has more than two preimages. The classes encompass known examples…

We study hermitian operators and isometries on spaces of vector-valued Lipschitz maps with the sum norm: $\|\cdot\|_{\infty}+L(\cdot)$. There are two main theorems in this paper. Firstly, we prove that every hermitian operator on…

Let $\varphi$ be a holomorphic self map of the bidisc that is Lipschitz on the closure. We show that the composition operator $C_{\varphi}$ is compact on the Bergman space if and only if $\varphi(\overline{\mathbb{D}^2})\cap…

Let $X$ and $Y$ be separable Banach spaces. Suppose $Y$ either has a shrinking basis or $Y$ is isomorphic to $C(2^\mathbb{N})$ and $A$ is a subset of weakly compact operators from $X$ to $Y$ which is analytic in the strong operator…

In this paper, we apply the Hausdorff measure of noncompactness to obtain the necessary and sufficient conditions for certain matrix operators on the Fibonacci difference sequence spaces l_{p}(F) and l_{infinite}(F) to be compact, where…

We give a characterization of the compact operators on a model space in terms of asymptotic Toeplitz operators.

We prove that if q is in (1,\infty), Y is a Banach space and T is a linear operator defined on the space of finite linear combinations of (1,q)-atoms in R^n which is uniformly bounded on (1,q)-atoms, then T admits a unique continuous…

Let $X$ be a separable Banach space, $Y$ be a Banach space and $\Lambda$ be a subset of the dual group of a given compact metrizable abelian group. We prove that if $X^*$ and $Y$ have the type I-$\Lambda$-RNP (resp. type II-$\Lambda$-RNP)…

We study compactness along orthonormal (disjoint bounded) sequences for operators from Hilbert spaces (Banach lattices) to Frechet spaces.

If $M$ is a compact smooth manifold and $X$ is a compact metric space, the Sobolev space $W^{1,p}(M,X)$ is defined through an isometric embedding of $X$ into a Banach space. We prove that the answer to the question whether Lipschitz…

We introduce notions of compactness and weak compactness for multilinear maps from a product of normed spaces to a normed space, and prove some general results about these notions. We then consider linear maps $T:A\to B$ between Banach…

There are two main aims of the paper. The first one is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second one is to extend the criterion for the…

We show that there are compact linear operators on Banach spaces which cannot be approximated by numerical radius attaining operators.