Quantification of Pełczyński's property (V)

Abstract

A Banach space has Pełczyński's property (V) if for every Banach space every unconditionally converging operator ∶ → is weakly compact. In 1962, Aleksander Pełczyński showed that ( ) spaces for a compact Hausdorff space enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about ( ) spaces and generalize it. Finally, we study the relationship of several properties of operators including weak compactness and unconditional convergence, and using the results obtained we establish a relation between quantitative versions of the property (V) and quantitative versions of other well known properties of Banach spaces. K E Y W O R D S Measures of weak non-compactness, Pełczyński's property (V), quantitative Pełczyński's property (V), unconditionally converging operators M S C ( 2 0 1 0 ) 46B04, 47B10, 46E15

Pełczyński's property () of order p and its quantification

“…For example, quantitative versions of Krein's theorem were studied in , quantitative versions of Eberlein–S̆mulyan and Gantmacher theorems were investigated in , a quantitative version of James' compactness theorem in , quantifications of weak sequential continuity and of the Schur property in and , quantitative Dunford–Pettis property in , quantification of the Banach–Saks property in , etc. H. Krulišová introduced several possibilities of quantifying Pełczyński's property ( V ). She introduced a quantity measuring to what extent a bounded subset K of fails to be a ( V )‐set and prove a quantitative version of the Pełczyński's characterization of Pełczyński's property ( V ).…”

Quantitative Bessaga–Pełczyński property and quantitative Rosenthal property

“…A well‐known characterization of unconditionally converging operators due to A. Pełczyński (also see [, Exercise 8, p. 54]) is that an operator is unconditionally converging if and only if T does not fix a c 0 ‐copy, that is, there is no subspace M of X isomorphic to c 0 such that the restriction of T to M is an isomorphism. To quantify this characterization, H. Krulišová introduced a quantity which measures the failure of the condition that T does not fix a copy of c 0 . For an operator , H. Krulišová sets …”

“…54]) is that an operator ∶ → is unconditionally converging if and only if does not fix a 0 -copy, that is, there is no subspace of isomorphic to 0 such that the restriction | of to is an isomorphism. To quantify this characterization, H. Krulišová [22] introduced a quantity 0 which measures the failure of the condition that does not fix a copy of 0 . For an operator ∶ → , H. Krulišová sets…”

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

“…Pe lczyński's property (V) and its quantification. Let us recall some essential definitions and facts (explained in more detail in [8] with many comments). A series ∞ n=1 x n in a Banach space X is said to be • unconditionally convergent if the series ∞ n=1 t n x n converges whenever (t n ) is a bounded sequence of scalars, • weakly unconditionally Cauchy (wuC ) if for all x ′ ∈ X ′ the series A bounded linear operator T : X → Y between Banach spaces X and Y is called unconditionally converging if n T x n is an unconditionally convergent series in Y whenever n x n is a weakly unconditionally Cauchy series in X.…”

“…It is not clear whether C * -algebras have also the property (V q ) * ω . From [8,Theorem 4.1] it follows that the answer is affirmative for commutative C * -algebras. In fact we do not know any example of a Banach space with the property (V q ) but not (V q ) * ω .…”