On the class of disjoint limited completely continuous operators

We introduce and study new class of sets (almost L-limited sets). Also, we introduce new concept of property in Banach lattice (almost Gelfand–Phillips property) and we characterize this property using almost L-limited sets. On the other hand, we introduce the class of disjoint limited completely continuous operators which is a largest class than that of limited completely continuous operators, we characterize this class of operators and we study some of its properties.     

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.     

An exact Fatou lemma for Gelfand integrals: a characterization of the Fatou property

We establish an exact version of Fatou’s lemma for Gelfand integrals of functions and multifunctions in dual Banach spaces without any order structure, and under the saturation property on the underlying measure space. The necessity and sufficiency of saturation for the Fatou property is demonstrated. Our result has a direct application to the equilibrium existence result for saturated economies without convexity assumptions.     

Integral mappings between Banach spaces

We consider the classes of “Grothendieck-integral” (G-integral) and “Pietsch-integral” (P-integral) linear and multilinear operators (see definitions below), and we prove that a multilinear operator between Banach spaces is G-integral (resp. P-integral) if and only if its linearization is G-integral (resp. P-integral) on the injective tensor product of the spaces, together with some related results concerning certain canonically associated linear operators. As an application we give a new proof of a result on the Radon-Nikodym property of the dual of the injective tensor product of Banach spaces. Moreover, we give a simple proof of a characterization of the G-integral operators on C(K,X) spaces and we also give a partial characterization of P-integral operators on C(K,X) spaces.     

An extension of a Phillips's theorem to Banach algebras and application to the uniform continuity of strongly continuous semigroups

In this work we present an extension to arbitrary unital Banach algebras of a result due to Phillips [R.S. Phillips, Spectral theory of semigroups of linear operators, Trans. Amer. Math. Soc. 71 (1951) 393-415] (Theorem 1.1) which provides sufficient conditions assuring the uniform continuity of strongly continuous semigroups of linear operators. It implies that, when dealing with the algebra of bounded operators on a Banach space, the conditions of Phillips's theorem are also necessary. Moreover, it enables us to derive necessary and sufficient conditions in terms of essential spectra which guarantee the uniform continuity of strongly continuous semigroups. We close the paper by discussing the uniform continuity of strongly continuous groups (T(t))_t∈R acting on Banach spaces with separable duals such that, for each t∈R, the essential spectrum of T(t) is a finite set.     

Best Linear Approximation Methods for Some Classes of Analytic Functions on the Unit Disk

Considering Banach Hardy spaces and weighted Bergman spaces, we find the sharp values of the Bernstein, Kolmogorov, Gelfand, and linear n-widths for the classes of analytic functions on the unit disk whose moduli of continuity of the rth derivatives averaged with weight are majorized by a given function satisfying some constraints.

     

On Interpolation Spaces with the Gelfand-Phillips Property

In this paper we study interpolation spaces generated by some interpolation functors. We show that under some conditions for Banach couples X and Y the spaces dual to the orbits of elements are Gelfand—Philips spaces. Consequently, the ideal of nuclear operators from X to Y contains a copy of l¹. We give also an interpolation theorem for limited operators.     

Rethinking polyhedrality for lindenstrauss spaces

We present a Lindenstrauss space with an extreme point that does not contain a subspace linearly isometric to c. This example disproves a result stated by Zippin in a paper published in 1969 and it shows that some classical characterizations of polyhedral Lindenstrauss spaces, based on Zippin’s result, are false, whereas some others remain unproven; then we provide a correct proof for those characterizations. Finally, we also disprove a characterization of polyhedral Lindenstrauss spaces given by Lazar, in terms of the compact norm-preserving extension of compact operators, and we give an equivalent condition for a Banach space X to satisfy this property.     

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 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.     

Some new rich subspaces of C. Applications

In this paper, we are interested in a class of subspaces of C, introduced by Bourgain [Studia Math. 77 (1984) 245-253]. Wojtaszczyk called them rich in his monograph [Banach Spaces for Analysts, Cambridge Univ. Press, 1991]. We give some new examples of such spaces: this allows us to recover previous results of Godefroy-Saab and Kysliakov on spaces with reflexive annihilator in a very simple way. We construct some other examples of rich spaces, hence having property (V) of Pe?czyński and Dunford-Pettis property. We also recover the results due to Bourgain and Saccone saying that spaces of uniformly convergent Fourier series share these properties, by only using the main result of [Studia Math. 77 (1984) 245-253] and some very elementary arguments. We generalize too these results.     

Factoring weakly compact operators

The main result is that every weakly compact operator between Banach spaces factors through a reflexive Banach space. Applications of the result and technique of proof include new results (e.g., separable conjugate spaces embed isomorphically in spaces with boundedly complete bases; convex weakly compact sets are affinely homeomorphic to sets in a reflexive space) and simple proofs of known results (e.g., there is a reflexive space failing the Banach-Saks property; if X is separable, then $\text{X =}\text{Z}{}^7\text{Z}$ for some Z; there is a separable space which does not contain l₁ whose dual is nonseparable).     

On the banach spaces with the property (V *) of Pelczynski. — II

Summary We present a converse of a result contained in our paper «On the Banach spaces with the property (V^*) of Pelczynski» so obtaining a characterization of that family of Banach spaces. Hence some extensions of other results from that note are presented. At the end we study property (V^*) in spaces of operators with compact range.The results of this paper were obtained during the permanence of the author at Kent State University, Kent, Ohio 44242, U.S.A. thanks to a fellowship of the Italian C.N.R.     

On quantification of weak sequential completeness

We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in L-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space X with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.     

On a generalized Mazur-Ulam question: Extension of isometries between unit spheres of Banach spaces

We call a Banach space X admitting the Mazur-Ulam property (MUP) provided that for any Banach space Y, if f is an onto isometry between the two unit spheres of X and Y, then it is the restriction of a linear isometry between the two spaces. A generalized Mazur-Ulam question is whether every Banach space admits the MUP. In this paper, we show first that the question has an affirmative answer for a general class of Banach spaces, namely, somewhere-flat spaces. As their immediate consequences, we obtain on the one hand that the question has an approximately positive answer: Given ε>0, every Banach space X admits a (1+ε)-equivalent norm such that X has the MUP; on the other hand, polyhedral spaces, CL-spaces admitting a smooth point (in particular, separable CL-spaces) have the MUP.     

Some remarks on James-Schreier spaces

The James-Schreier spaces Vp, where 1?p<∞, were recently introduced by Bird and Laustsen (in press) [5] as an amalgamation of James' quasi-reflexive Banach space on the one hand and Schreier's Banach space giving a counterexample to the Banach-Saks property on the other. The purpose of this note is to answer some questions left open in Bird and Laustsen (in press) [5]. Specifically, we prove that (i) the standard Schauder basis for the first James-Schreier space V₁ is shrinking, and (ii) any two Schreier or James-Schreier spaces with distinct indices are non-isomorphic. The former of these results implies that V₁ does not have Pe?czyński's property (u) and hence does not embed in any Banach space with an unconditional Schauder basis.     

The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ? starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ? starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ? starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.     

Some remarks on the Krein-Milman property and lexicographic maxima

A Banach space has thelexicographic property if each bounded closed convex subsetC is the closed convex hull of the lexicographic maxima ofC. The relationships between this property and the Radon-Nikodym and Krein-Milman properties are provided. A example inl ¹ contrasts these concepts. The main result states that a Banach space which is anr-space has the lexicographic property. Several questions relating to these topics and also flat spaces are posed.     

Borel properties of linear operators

Given an injective bounded linear operator T:X→Y between Banach spaces, we study the Borel measurability of the inverse map T−1:TX→X. A remarkable result of Saint-Raymond (Ann. Inst. Fourier (Grenoble) 26 (1976) 211-256) states that if X is separable, then the Borel class of T⁻¹ is α if, and only if, X∗ is the αth iterated sequential weak∗-closure of T∗Y∗ for some countable ordinal α. We show that Saint-Raymond's result holds with minor changes for arbitrary Banach spaces if we assume that T has certain property named co-σ-discreteness after Hansell (Proc. London Math. Soc. 28 (1974) 683-699). As an application, we show that the Borel class of the inverse of a co-σ-discrete operator T can be estimated by the image of the unit ball or the restrictions of T to separable subspaces of X. Our results apply naturally when X is a WCD Banach space since in this case any injective bounded linear operator defined on X is automatically co-σ-discrete.     

Operator martingale decompositions and the Radon-Nikodým property in Banach spaces

We consider submartingales and uniform amarts of maps acting between a Banach lattice and a Banach lattice or a Banach space. In this measure-free setting of martingale theory, it is known that a Banach space Y has the Radon-Nikodým property if and only if every uniformly norm bounded martingale defined on the Chaney-Schaefer l-tensor product , where E is a suitable Banach lattice, is norm convergent. We present applications of this result. Firstly, an analogues characterization for Banach lattices Y with the Radon-Nikodým property is given in terms of a suitable set of submartingales (supermartingales) on . Secondly, we derive a Riesz decomposition for uniform amarts of maps acting between a Banach lattice and a Banach space. This result is used to characterize Banach spaces with the Radon-Nikodým property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent. In the case 1<p<∞, our results yield Lp(μ,Y)-space analogues of some of the well-known results on uniform amarts in L¹(μ,Y)-spaces.