Related papers: Operator ideals on non-commutative function spaces

We call a bounded linear operator acting between Banach spaces weakly compactly generated ($\mathsf{WCG}$ for short) if its range is contained in a weakly compactly generated subspace of its codomain. This notion simultaneously generalises…

In this paper, we present some necessary and sufficient conditions for semi-compact operators being almost L-weakly compact (resp. almost M-weakly compact) and the converse. Mainly, we prove that if $X$ is a nonzero Banach space, then every…

We characterize classes of linear maps between operator spaces $E$, $F$ which factorize through maps arising in a natural manner via the Pisier vector-valued non-commutative $L^p$ spaces $S_p[E^*]$ based on the Schatten classes on the…

This paper contains the following results: a) Suppose that X is a non-trivial Banach space and L is a non-empty locally compact Hausdorff space without any isolated points. Then each linear operator T: C_{0}(L,X)\to C_{0}(L,X), whose range…

We give sufficient conditions on an asymptotic $\ell_p$ (for $1 < p < \infty$) Banach space which ensure the space admits an operator which is not a compact perturbation of a multiple of the identity. These conditions imply the existence of…

Given an operator ideal I, a Banach space E has the I-approximation property if operators on E can be uniformly approximated on compact subsets of E by operators belonging to I. In this paper the I- approximation property is studied in…

A Banach space $X$ is said to have the Daugavet property if every operator $T: X\to X$ of rank~$1$ satisfies $\|Id+T\| = 1+\|T\|$. We show that then every weakly compact operator satisfies this equation as well and that $X$ contains a copy…

In this paper we present part I of nonlinear operator ideals theory between metric spaces and Banach spaces. Building upon the definition of operator ideal between arbitrary Banach spaces of A. Pietsch we pose three types of nonlinear…

We prove that every lattice homomorphism acting on a Banach space $\mathcal{X}$ with the lattice structure given by an unconditional basis has a non-trivial closed invariant subspace. In fact, it has a non-trivial closed invariant ideal,…

Let $X$, $Y$ be Banach spaces and let $\mathcal{L}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on $\mathcal{L}(X,Y)$ and apply this theory to obtain commutator estimates…

Let $\lambda$ be an infinite cardinal number and let $\ell_\infty^c(\lambda)$ denote the subspace of $\ell_\infty(\lambda)$ consisting of all functions that assume at most countably many non-zero values. We classify all infinite dimensional…

We propose a unifying approach to many approximation properties studied in the literature from the 1930s up to our days. To do so, we say that a Banach space E has the (I,J,{\tau})-approximation property if E-valued operators belonging to…

Suppose $\Cal J$ is a two-sided quasi-Banach ideal of compact operators on a separable infinite-dimensional Hilbert space $\Cal H$. We show that an operator $T\in\Cal J$ can be expressed as finite linear combination of commutators $[A,B]$…

The theory of M-ideals and multiplier mappings of Banach spaces naturally generalizes to left (or right) M-ideals and multiplier mappings of operator spaces. These subspaces and mappings are intrinsically characterized in terms of the…

We prove that if T is a strictly singular 1-1 operator defined on an infinite dimensional Banach space X, then for every infinite dimensional subspace Y of X there exists an infinite dimensional subspace Z of Y such that Z contains orbits…

In this paper, we establish some results about the singular points of certain non-monotone potential operators. Here is a sample: If $X$ is an infinite-dimensional reflexive real Banach space and if $T:X\to X^*$ is a non-monotone, closed,…

We consider majorization problems in the non-commutative setting. More specifically, suppose $E$ and $F$ are ordered normed spaces (not necessarily lattices), and $0 \leq T \leq S :E \to F$. If $S$ belongs to a certain ideal (for instance,…

Let $\mathcal H$ be an infinite-dimensional Hilbert space, and let $\mathcal B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^*$-algebra of bounded (respectively, compact) linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a…

We introduce and develop the notion of hyper-ideals of multilinear operators between Banach spaces. While the well studied notion of ideals of multilinear operators (multi-ideals) relies on the composition with linear operators, the notion…

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)…