An operator T : E → X between a Banach lattice E and a Banach space X is called b-weakly compact if T (B) is relatively weakly compact for each b-bounded set B in E. We characterize b-weakly compact operators among o-weakly compact operators. We show summing operators are b-weakly compact and discuss relation between Dunford–Pettis and b-weakly compact operators. We give necessary conditions fo...

Let  $left{a_alpharight}_{alphain I}$ be a bounded net in a Banach algebra $A$ and $varphi$ a nonzero multiplicative linear functional on $A$. In this paper, we deal with the problem of when $|aa_alpha-varphi(a)a_alpha|to0$ uniformly for all $a$ in weakly compact subsets of $A$. We show that Banach algebras associated to locally compact groups such  as Segal algebras and $L^1$-algebras are resp...

A Banach space E has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space P(kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric k-fold projective tensor product of E (i.e., the predual of P...

In this note, we aim to present some properties of the space of all weakly fuzzy bounded linear operators, with the Bag and Samanta’s operator norm on Felbin’s-type fuzzy normed spaces. In particular, the completeness of this space is studied. By some counterexamples, it is shown that the inverse mapping theorem and the Banach-Steinhaus’s theorem, are not valid for this fuzzy setting. Also...

in this note, we aim to present some properties of the space of all weakly fuzzy bounded linear operators, with the bag and samanta’s operator norm on felbin’s-type fuzzy normed spaces. in particular, the completeness of this space is studied. by some counterexamples, it is shown that the inverse mapping theorem and the banach-steinhaus’s theorem, are not valid for this fuzzy setting. also...

Let T : C(S) → C(S) be a bounded linear operator. We present a necessary and sufficient condition for the so-called Daugavet equation ‖Id+ T ‖ = 1 + ‖T ‖ to hold, and we apply it to weakly compact operators and to operators factoring through c0. Thus we obtain very simple proofs of results by Foiaş, Singer, Pe lczyński, Holub and others. If E is a real Banach space, let us say that an operator ...

A bounded linear operator between Banach spaces is called completely continuous if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators from L1 into an arbitrary Banach space, namely, the operator from L1 into `∞ defined by T0(f) = Z rnf dμ n≥0 , where rn is the nth Rademacher function. It is...

Letting E, F be Banach spaces, the main two results of this paper are the following: (1) If every (linear bounded) operator E → F is unconditionally converging, then every polynomial from E to F is unconditionally converging (definition as in the linear case). (2) If E has the Dunford-Pettis property and every operator E → F is weakly compact, then every k-linear mapping from E into F takes wea...