Related papers: Genericity and Universality for Operator Ideals

The powerful concept of an operator ideal on the class of all Banach spaces makes sense in the real and in the complex case. In both settings we may, for example, consider compact, nuclear, or $2$--summing operators, where the definitions…

A bounded linear operator between Banach spaces is called {\it completely continuous} if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous…

Suppose $X$ and $Y$ are Banach spaces, and ${\mathcal{I}}$, ${\mathcal{J}}$ are operator ideals (for instance, the ideals of strictly singular, weakly compact, or compact operators). Under what conditions does the inclusion…

In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compact, weakly compact, completely continuous, Banach-Saks and weakly Banach-Saks…

We use the notion of $\A$-compact sets, which are determined by a Banach operator ideal $\A$, to show that most classic results of certain approximation properties and several Banach operator ideals can be systematically studied under this…

A bounded operator on a real or complex separable infinite-dimensional Banach space $Z$ is universal in the sense of Glasner and Weiss if for every invertible ergodic measure-preserving transformation $T$ of a standard Lebesgue probability…

We develop general techniques and present an approach to solve the problem of constructing a maximal Banach ideal $({\frak A},{\bf A)}$ which does not satisfy a transfer of the norm estimation in the principle of local reflexivity to its…

In addition to Pisier's counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which {\it{are accessible}}. The first step is implied by the observation that a "good behaviour" of trace…

Let $X_1, \ldots, X_n,Y$ be classes of Banach spaces-valued sequences. An $n$-linear operator $A$ between Banach spaces belongs to the ideal of $(X_1, \ldots, X_n;Y)$-summing multilinear operators if $(A(x_j^1, \ldots, x_j^n))_{j=1}^\infty$…

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 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…

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 $[A,B]$…

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…

We compare two types of universal operators constructed relatively recently by Cabello S\'anchez, and the authors. The first operator $\Omega$ acts on the Gurarii space, while the second one $P_S$ has values in a fixed separable Banach…

In this paper we study two types of collections of operators on a Banach space on the subject of forming operator ideals. One of the types allows us to construct an uncountable chain of closed ideals in each of the operator algebras…

In this article, we give an abstract characterization of the ``identity'' of an operator space $V$ by looking at a quantity $n_{cb}(V,u)$ which is defined in analogue to a well-known quantity in Banach space theory. More precisely, we show…

A linear operator $U$ acting boundedly on an infinite-dimensional separable complex Hilbert space $H$ is universal if every linear bounded operator acting on $H$ is similar to a scalar multiple of a restriction of $U$ to one of its…

Generalizing classical results of the theory of absolutely summing operators, in this paper we characterize the duals of a quite large class of Banach operator ideals defined or characterized by the transformation of vector-valued…

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…

Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space, $\mathcal{B}(\mathcal{H})$ the algebra of bounded linear operators acting on $\mathcal{H}$ and $\mathcal{J}$ a proper two-sided ideal of…