Banach Operator Ideals Research Articles

On some measures of non-compactness associated to Banach operator ideals

We study two variants of measures of non-compactness of operators associated to a Banach operator ideal in the sense of Pietsch. These measures are motivated by the notions of surjective-ideal-compactness and injective-ideal-compactness, defined respectively by Carl and Stephani and by Stephani. Interpolation results on these measures in the cases of Banach couples generated by a single Banach space are given. As an application, we obtain interpolation theorems on p -compact operators and quasi p -nuclear operators.

Quotient algebras of Banach operator ideals related to non-classical approximation properties

We investigate the quotient algebra AXI:=I(X)/F(X)‾||⋅||I for Banach operator ideals I contained in the ideal of the compact operators, where X is a Banach space that fails the I-approximation property. The main results concern the nilpotent quotient algebras AXQNp and AXSKp for the quasi p-nuclear operators QNp and the Sinha-Karn p-compact operators SKp. The results include the following: (i) if X has cotype 2, then AXQNp={0} for every p≥1; (ii) if X⁎ has cotype 2, then AXSKp={0} for every p≥1; (iii) the exact upper bound of the index of nilpotency of AXQNp and AXSKp for p≠2 is max⁡{2,⌈p/2⌉}, where ⌈p/2⌉ denotes the smallest n∈N such that n≥p/2; (iv) for every p>2 there is a closed subspace X⊂c0 such that both AXQNp and AXSKp contain a countably infinite decreasing chain of closed ideals. In addition, our methods yield a closed subspace X⊂c0 such that the compact-by-approximable algebra AX=K(X)/A(X) contains two incomparable countably infinite chains of nilpotent closed ideals.

Duality theory for generalized summing linear operators

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 sequences.

Closed ideals in the algebra of compact-by-approximable operators

We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra AX:=K(X)/A(X) on Banach spaces X failing the approximation property. The examples include the following: (i) if X has cotype 2, Y has type 2, AX≠{0} and AY≠{0}, then AX⊕Y has at least 2 closed ideals, (ii) there are closed subspaces X⊂ℓp for 4<p<∞ and X⊂c0 such that AX contains a non-trivial closed ideal, (iii) there is a Banach space Z such that AZ contains an uncountable lattice of closed ideals having the reverse order structure of the power set of the natural numbers. Some of our results involve non-classical approximation properties associated to various Banach operator ideals. We also discuss the existence of compact non-approximable operators X→Y, where X⊂ℓp and Y⊂ℓq are closed subspaces for p≠q.

Injective Dual Banach Spaces and Operator Ideals

We give characterizations of Banach spaces with $$\lambda $$ -injective biduals in terms of operators with an integral representation. These results may be thought of as a certain refinement of Lindenstrauss’ extension theorems from his 1964 memoir. We also obtain dual results characterizing Banach spaces with $$\lambda $$ -injective duals in terms of factorization of operators through $$L_1(\mu )$$ or $$\ell _1$$ .

Banach Compactness and Banach Nuclear Operators

In this paper, we introduce the notion of (uniformly weakly) Banach-compact sets, (uniformly weakly) Banach-compact operators and (uniformly weakly) Banach-nuclear operators which generalize p-compact sets, p-compact operators and p-nuclear operators, respectively. Fundamental properties are investigated. Factorizations and duality theorems are given. Injective and surjective hulls are used to show the spaces of (uniformly weakly) Banach-compact operators and (uniformly weakly) Banach-nuclear operators are quasi Banach operator ideals.

Operator p-compact mappings

We introduce the classes of operator p-compact mappings and completely right p-nuclear operators, which are natural extensions to the operator space framework of their corresponding Banach operator ideals. We relate these two classes, define natural operator space structures and study several properties of these ideals. We show that the class of operator ∞-compact mappings in fact coincides with a notion already introduced by Webster in the nineties (in a very different language). This allows us to provide an operator space structure to Webster's class.

Galois connection between Lipschitz and linear operator ideals and minimal Lipschitz operator ideals

We establish a relation between Lipschitz operator ideals and linear operator ideals, which fits in the framework of Galois connection between lattices. We use this relationship to give a criterion which allows us to recognize when a Banach Lipschitz operator ideal is of composition type or not. Also, we introduce the concept of minimal Banach Lipschitz operator ideal, which have analogous properties to minimal Banach operator ideals. We characterize minimal Banach Lipschitz operator ideals which are of composition type and present examples which are not of this class.

On null sequences for Banach operator ideals, trace duality and approximation properties

Fil: Lassalle, Silvia Beatriz. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Investigaciones Matematicas ; Argentina

The approximation properties determined by operator ideals

We introduce the notion of the right approximation property with respect to an operator ideal A and solve the duality problem for the approximation property with respect to an operator ideal A, that is, a Banach space X has the approximation property with respect to A d whenever X* has the right approximation property with respect to an operator ideal A. The notions of the left bounded approximation property and the left weak bounded approximation property for a Banach operator ideal are introduced and new symmetric results are obtained. Finally, the notions of the p-compact sets and the p-approximation property are extended to arbitrary Banach operator ideals. Known results of the approximation property with respect to an operator ideal and the p-approximation property are generalized.

Generalized Orlicz-Lorentz sequence spaces and corresponding operator ideals

Abstract In this paper we introduce generalized or vector-valued Orlicz-Lorentz sequence spaces l p,q,M(X) on Banach space X with the help of an Orlicz function M and for different positive indices p and q. We study their structural properties and investigate cross and topological duals of these spaces. Moreover these spaces are generalizations of vector-valued Orlicz sequence spaces l M(X) for p = q and also Lorentz sequence spaces for M(x) = x q for q ≥ 1. Lastly we prove that the operator ideals defined with the help of scalar valued sequence spaces l p,q,M and additive s-numbers are quasi-Banach operator ideals for p &lt; q and Banach operator ideals for p ≥ q. The results of this paper are more general than the work of earlier mathematicians, say A. Pietsch, M. Kato, L. R. Acharya, etc.

The Banach ideal of [formula omitted]-compact operators and related approximation properties

We use the notion of A-compact sets (determined by an operator ideal A), introduced by Carl and Stephani (1984), to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework. For Banach operator ideals A, we introduce a way to measure the size of A-compact sets and use it to give a norm on KA, the ideal of A-compact operators. Then, we study two types of approximation properties determined by A-compact sets. We focus our attention on an approximation property which makes use of the norm defined on KA. This notion fits the definition of the A-approximation property, recently introduced by Oja (2012), with KA instead of A. We exemplify the power of the Carl–Stephani theory and the geometric structure introduced here by appealing to some recent developments on p-compactness.

Interpolative construction and factorization of operators

Banach operator ideals generated by interpolative construction applied to p-summing operators are studied. These ideals are described in terms of factorization through abstract interpolation Lorentz spaces. Relationships between Banach ideals determined by Orlicz sequence spaces are shown and a variant of the Pisier factorization theorem for (p,1)-summing operators from C(K)-spaces is proved. Applications to Schatten classes are given. It is also shown that certain known results on (q,p)-concave operators from Banach lattices can be lifted to a class of generalized concave operators.

A particular infinite matrix and related norms

We investigate norms of vectors and operators obtained from the infinite scalar matrix where thereby improving and complementing results of [5]. For any choice of and , U gives rise to a bounded linear operator which enjoys compactness properties close to nuclearity. Whereas we cannot characterize for which (p, q) the operator is nuclear, we will show that frequently this is the case, and that for certain Banach operator ideals, the corresponding ideal norms for and/or related operators coincide with the usual operator norm. On the other hand for example, is easily seen to be a Hilbert-Schmidt operator, but neither its norm nor its Hilbert-Schmidt norm are explicitly known. However, at least for the Hilbert-Schmidt norm, reasonably efficient approximation schemes are available.

Interpolative construction and the generalized cotype of abstract Lorentz spaces

The Banach operator ideals generated by an interpolative construction depending on concave functions are studied. These ideals are described in terms of factorization through abstract interpolation Lorentz spaces. The abstract notion of Rademacher type and cotype for operators between Banach spaces is introduced. It is shown that abstract interpolation Lorentz spaces that appeared in the factorization theorem are of the generalized Rademacher cotype determined by Orlicz sequence spaces.

Duality of the distance to closed operator ideals

Special operator-ideal approximation properties (APs) of Banach spaces are employed to solve the problem of whether the distance functions S ↦ dist(S*, I(F*, E*)) and S ↦ dist(S, I*(E, F)) are uniformly comparable in each space L(E, F) of bounded linear operators. Here, I*(E, F) = {S ∈ L(E, F) : S* ∈ I(F*, E*)} stands for the adjoint ideal of the closed operator ideal I for Banach spaces E and F. Counterexamples are obtained for many classical surjective or injective Banach operator ideals I by solving two resulting ‘asymmetry’ problems for these operator-ideal APs.

On the continuity of scales of some operator ideals

The continuity of p-absolutely summing and p-nuclear norms in the corresponding scales of Banach operator ideals is studied. Bibliography: 17 titles.

Some Operator Ideals in Non-Commutative Functional Analysis

We study classes of linear maps between operator spaces E and F which factorize through maps arising in a natural manner by the Pisier vector-valued non-commutative L^p -spaces S_p[E] based on the Schatten classes on the separable Hilbert space \ell ^2 . These classes of maps, firstly introduced in [28] and called p-nuclear maps, can be viewed as Banach operator ideals in the category of operator spaces, that is in non-commutative (quantized) functional analysis. We also discuss some applications to the split property for inclusions of W* -algebras such as those describing the physical observables in Quantum Field Theory.

Weak sequential completeness of Banach operator ideals

Weak sequential completeness of Banach operator ideals

Remarks on the classical Banach operator ideals

Remarks on the classical Banach operator ideals