Fredholm theory forp-adic locally convex spaces

Summary The authors develop the Fredholm theory for semi-compact operators on non-archimedean locally convex spaces. This theory coincides with Schikhof's Fredholm theory for compact operators on Banach spaces which fails for non-complete normed spaces.     

On the weak topology of Banach spaces over non-archimedean fields

It is known that within metric spaces analyticity and K-analyticity are equivalent concepts. It is known also that non-separable weakly compactly generated (shortly WCG) Banach spaces over R or C provide concrete examples of weakly K-analytic spaces which are not weakly analytic. We study the case which totally differs from the above one. A general theorem is provided which shows that a Banach space E over a locally compact non-archimedean non-trivially valued field is weakly Lindelöf iff E is separable iff E is WCG iff E is weakly web-compact (in the sense of Orihuela). This provides a non-archimedean version of a remarkable Amir-Lindenstrauss theorem.     

Géométrie des espaces de Banach ayant des approximations de l'identité contractantes

Let a, c ≥ 0 and let B be a compact set of scalars. We introduce property M^* (a, B, c) of Banach spaces X which is a geometric property of Banach spaces generalizing property (M^*) due to Kalton. Using M^*(a, B, c) with max ¦B¦ + c > 1, we characterize intrinsically a large class of shrinking approximations of the identity, including those related to M-, u-, and h-ideals of compact operators. We also show that the existence of these approximations of the identity is separably determined. As an application, we study ideals of compact and approximable operators. In particular, this provides an alternative unified and easier approach to the theories of M-, u-, and h-ideals of compact operators.     

Linearization of multipolynomials and applications

We prove that every multipolynomial between Banach spaces is the composition of a canonical multipolynomial with a linear operator, and that this correspondence establishes an isometric isomorphism between the spaces of multipolynomials and linear operators. Applications to composition ideals of multipolynomials and to multipolynomials that are of finite rank, approximable, compact, and weakly compact are provided.     

Bishop–Phelps–Bollobás property for certain spaces of operators

We characterize the Banach spaces Y   for which certain subspaces of operators from _L1(μ)$L_1(\mu )$ into Y have the Bishop–Phelps–Bollobás property in terms of a geometric property of Y, namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain that certain ideals of Asplund operators satisfy the Bishop–Phelps–Bollobás property.     

Perturbation theory of p-adic Fredholm and semi-Fredholm operators

The main goal of this paper is to prove that Fredholm and semi-Fredholm operators between p-adic (or non-archimedean) Banach spaces, as well as the index of those that are Fredholm, are preserved when they are perturbed by a small operator. In this way we obtain the non-archimedean counterparts of some well-known results of classical Operator Theory. For non-spherically complete fields the classical techniques are no longer valid in the p-adic context, which forces us to seek a completely different way to attack the problem. The p-adic concept of orthogonality will be one of the key tools to get our purpose.     

Compact groups of positive operators on Banach lattices

In this paper, we study groups of positive operators on Banach lattices. If a certain factorization property holds for the elements of such a group, the group has a homomorphic image in the isometric positive operators which has the same invariant ideals as the original group. If the group is compact in the strong operator topology, it equals a group of isometric positive operators conjugated by a single central lattice automorphism, provided an additional technical assumption is satisfied, for which we have only examples. We obtain a characterization of positive representations of a group with compact image in the strong operator topology, and use this for normalized symmetric Banach sequence spaces to prove an ordered version of the decomposition theorem for unitary representations of compact groups. Applications concerning spaces of continuous functions are also considered.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

On the instability of non-semi-Fredholm operators under compact perturbations

The stability of several natural sets of the non-semi-Fredholm operators in a separable Hilbert space under compact perturbations studied by R. Bouldin. (The instability of non-semi-Fredholm operators under compact perturbations, J. Math. Anal. Appl.87 (1982), 632–638.) The aim of the present article is to study this problem in arbitrary Banach spaces. We also derive a curious characterization of separable Banach spaces.     

Dominated inessential operators

We consider the following three closed algebraic ideals of operators on a Banach lattice: compact, strictly singular, and inessential operators. Suppose that 0?A?B and B is compact or strictly singular. We show that, under certain assumptions, A (or some power of A) is inessential.     

Operator Ideals on Non-commutative Function Spaces

Suppose X and Y are Banach spaces, and \({{\mathcal{I}}}\) , \({{\mathcal{J}}}\) are operator ideals. compact operators). Under what conditions does the inclusion \({\mathcal{I}(X,Y) \subset \mathcal{J}(X,Y)}\) , or the equality \({\mathcal{I}(X,Y)\,=\,\mathcal{J}(X,Y)}\) , hold? We examine this question when \({\mathcal{I}, \mathcal{J}}\) are the ideals of Dunford–Pettis, strictly (co)singular, finitely strictly singular, inessential, or (weakly) compact operators, while X and Y are non-commutative function spaces. Since such spaces are ordered, we also address the same questions for positive parts of such ideals.     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

On Banach spaces with the Gelfand-Phillips property. II

We present some result of lifting of the Gelfand Phillips property from Banach spacesE andF to Banach spaces of compact operators and of Bochner integrable functions. Moreover we studyC(K) spaces possessing the same property. In the last section we prove some result concerning the so called three space problem for the Gelfand Phillips property too.     

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.     

An Extension of the Notion of Orthogonality to Banach Spaces

We extend the usual notion of orthogonality to Banach spaces. We show that the extension is quite rich in structure by establishing some of its main properties and consequences. Geometric characterizations and comparison results with other extensions are established. Also, we establish a characterization of compact operators on Banach spaces that admit orthonormal Schauder bases. Finally, we characterize orthogonality in the spaces l₂^p(C).     

Banach ideals of p-compact operators

The main theme of this paper is a study in some detail of Banach ideals of continuous linear operators between Banach spaces factoring compactly through l^p (1p<) or c_o, called p-compact and -compact operators respectively. Recently operators of these types have been studied in [4] within the framework of locally convex spaces which are dense subspaces of p-compact projective limits of Banach spaces. These ideals show close resemblance to the ideals of p-nuclear operators-for the case p= they coincide. Analogously to results of Grothendieck concerning continuous linear operators, we consider vector sequence spaces isometric isomorphic to certain spaces of compact linear operators. A representation theorem for p-compact operators is deduced and isometric properties of the ideal norm are treated. The paper also includes some remarks on unconditional convergence and related operator ideals and a representation for the complete -tensor product (1p<) is given.     

L-weakly and M-weakly compact operators and the centre

We extend known results concerning the centre of spaces of regular (resp. weakly compact or compact) operators between two Banach lattices to the setting of L-weakly compact and M-weakly compact operators. We also show that the L-weakly compact, M-weakly compact, and compact operators lying in the centre of a Banach lattice coincide.     

Weakly closed Jordan ideals in nest algebras on Banach spaces

We show that weakly closed Jordan ideals in nest algebras on Banach spaces are associative ideals. The decomposability of finite-rank operators in Jordan ideals and the commutants of bimodules are also investigated.     

Techniques to generate hyper-ideals of multilinear operators

The usual techniques to generate ideals of multilinear operators between Banach spaces fail in generating hyper-ideals in general. In this paper, we fill this gap by developing two techniques to generate hyper-ideals of multilinear operators. The techniques we develop generate new classes of multilinear operators and show that some important well-studied classes are Banach or p-Banach hyper-ideals.     

Hypercyclic convolution operators on Fréchet spaces of analytic functions

A result of Godefroy and Shapiro states that the convolution operators on the space of entire functions on Cn, which are not multiples of identity, are hypercyclic. Analogues of this result have appeared for some spaces of holomorphic functions on a Banach space. In this work, we define the space holomorphic functions associated to a sequence of spaces of polynomials and determine conditions on this sequence that assure hypercyclicity of convolution operators. Some known results come out as particular cases of this setting. We also consider holomorphic functions associated to minimal ideals of polynomials and to polynomials of the Schatten-von Neumann class.