The weak metric approximation property and ideals of operators

We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y.     

The approximation property in terms of the approximability of weak∗-weak continuous operators

By a well-known result of Grothendieck, a Banach space X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous compact operator T:X∗→Y can be uniformly approximated by finite rank operators from X⊗Y. We prove the following “metric” version of this criterion: X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous weakly compact operator T:X∗→Y can be approximated in the strong operator topology by operators of norm ?‖T‖ from X⊗Y. As application, easier alternative proofs are given for recent criteria of approximation property due to Lima, Nygaard and Oja.     

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pe?czyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.     

Geometry of spaces of compact operators

We introduce the notion of compactly locally reflexive Banach spaces and show that a Banach space X is compactly locally reflexive if and only if for all reflexive Banach spaces Y. We show that X ^* has the approximation property if and only if X has the approximation property and is compactly locally reflexive. The weak metric approximation property was recently introduced by Lima and Oja. We study two natural weak compact versions of this property. If X is compactly locally reflexive then these two properties coincide. We also show how these properties are related to the compact approximation property and the compact approximation property with conjugate operators for dual spaces.     

Three-space problems for the bounded compact approximation property

In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the BCAP, then L∞/X has the BCAP. We also show that X* has the λ-BCAP with conjugate operators if and only if the pair (X, Y) has the λ-BCAP for each finite codimensional subspace Y∈X. Let M be a closed subspace of X such that M⊥ is complemented in X*. If X has the (bounded) approximation property of order p, then M has the (bounded) approximation property of order p.     

Density of finite rank operators in the Banach space of p-compact operators

A Banach space X is said to have the kp-approximation property (kp-AP) if for every Banach space Y, the space F(Y,X) of finite rank operators is dense in the space Kp(Y,X) of p-compact operators endowed with its natural ideal norm kp. In this paper we study this notion that has been previously treated by Sinha and Karn (2002) in [15]. As application, the kp-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi p-nuclear operators for the p-summing norm. This allows to obtain a relation between the kp-AP and Saphar's approximation property. As another application, the kp-AP is characterized in terms of a trace condition. Finally, we relate the kp-AP to the (p,p)-approximation property introduced in Sinha and Karn (2002) [15] for subspaces of Lp(μ)-spaces.     

The Bishop-Phelps-Bollobás theorem for operators

We prove the Bishop-Phelps-Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollobás theorem holds for operators from ?₁ into Y. Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollobás theorem holds when the range space is finite-dimensional, an L₁(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.     

Isometric embeddings of compact spaces into Banach spaces

We show the existence of a compact metric space K such that whenever K embeds isometrically into a Banach space Y, then any separable Banach space is linearly isometric to a subspace of Y. We also address the following related question: if a Banach space Y contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space X, does it necessarily contain a subspace isometric to X? We answer positively this question when X is a polyhedral finite-dimensional space, c₀ or ?₁.     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.     

Dual problems for weak and quasi approximation properties

It is shown that for the separable dual X^∗ of a Banach space X if X^∗ has the weak approximation property, then X has the metric quasi approximation property. Using this it is shown that for the separable dual X^∗ of a Banach space X the quasi approximation property and metric quasi approximation property are inherited from X^∗ to X and for a separable and reflexive Banach space X, X having the weak approximation property, bounded weak approximation property, quasi approximation property, metric weak approximation property, and metric quasi approximation property are equivalent. Also it is shown that the weak approximation property, bounded weak approximation property, and quasi approximation property are not inherited from a Banach space X to X^∗.     

The <Emphasis Type="Italic">p</Emphasis>-Gelfand–Phillips property in spaces of operators and Dunford–Pettis like sets

The p-Gelfand–Phillips property (1 \({\leq}\) p < ∞) is studied in spaces of operators. Dunford–Pettis type like sets are studied in Banach spaces. We discuss Banach spaces X with the property that every p-convergent operator T:X \({\rightarrow}\) Y is weakly compact, for every Banach space Y.     

C*-algebras have a quantitative version of Pełczyński’s property (V)

A Banach space X has Pe?czyński’s property (V) if for every Banach space Y every unconditionally converging operator T: X → Y is weakly compact. H.Pfitzner proved that C*-algebras have Pe?czyński’s property (V). In the preprint (Kruli?ová, (2015)) the author explores possible quantifications of the property (V) and shows that C(K) spaces for a compact Hausdorff space K enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover, we prove that in dual Banach spaces a quantitative version of the property (V) implies a quantitative version of the Grothendieck property.     

The Grothendieck property for injective tensor products of Banach spaces

Let X be a Banach space with the Grothendieck property, Y a reflexive Banach space, and let X ⊗̌_ɛ Y be the injective tensor product of X and Y.

The asymptotically commuting bounded approximation property of Banach spaces

We introduce and study the asymptotically commuting bounded approximation property of Banach spaces. This property is, e.g., enjoyed by any dual space with the bounded approximation property. The principal result is the following: if a Banach space X has the asymptotically λ-commuting bounded approximation property, then X is saturated with locally λ-complemented separable subspaces enjoying the λ-commuting bounded approximation property.     

On dual and three space problems for the compact approximation property

We introduce the properties W^∗D and BW^∗D for the dual space of a Banach space. And then solve the dual problem for the compact approximation property (CAP): if X^∗ has the CAP and the W^∗D, then X has the CAP. Also, we solve the three space problem for the CAP: for example, if M is a closed subspace of a Banach space such that M^⊥ is complemented in X^∗ and X^∗ has the W^∗D, then X has the CAP whenever X/M has the CAP and M has the bounded CAP. Corresponding problems for the bounded compact approximation property are also addressed.     

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

We show that a Banach space E has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on E can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.     

On relations between weak approximation properties and their inheritances to subspaces

It is shown that for the separable dual X^∗ of a Banach space X, if X^∗ has the weak approximation property, then X^∗ has the metric weak approximation property. We introduce the properties W^∗D and MW^∗D for Banach spaces. Suppose that M is a closed subspace of a Banach space X such that M^⊥ is complemented in the dual space X^∗, where for all m∈M}. Then it is shown that if a Banach space X has the weak approximation property and W^∗D (respectively, metric weak approximation property and MW^∗D), then M has the weak approximation property (respectively, bounded weak approximation property).     

Compactness in

This paper is concerned with compactness for some topologies on the collection of bounded linear operators on Banach spaces. New versions of the Eberlein–Šmulian theorem and Day's lemma in the collection are established. Also we obtain a partial solution of the dual problem for the quasi approximation property, that is, it is shown that for a Banach space X if X^** is separable and X^* has the quasi approximation property, then X has the quasi approximation property.     

Little Grothendieck's theorem for sublinear operators

Let SB(X,Y) be the set of the bounded sublinear operators from a Banach space X into a Banach lattice Y. Consider π₂(X,Y) the set of 2-summing sublinear operators. We study in this paper a variation of Grothendieck's theorem in the sublinear operators case. We prove under some conditions that every operator in SB(C(K),H) is in π₂(C(K),H) for any compact K and any Hilbert H. In the noncommutative case the problem is still open.     

Compactness and collective compactness in spaces of compact operators

This is a study of compactness in (a) spaces K_b(X, Y) of compact linear operators, (b) injective tensor products $\text{X}\text{}\text{?}\text{bo}{}_{\mathrm{?}}\text{Y}$, and (c) spaces L_c(X, Y) of continuous linear operators, and its various relationships with equicontinuity and collective compactness. Among the applications is a result on factoring compact sets of compact operators compactly and uniformly through one and the same reflexive Banach space.