In this paper, we introduce and study new concepts of b-L-weakly and order M-weakly compact operators. As consequences, we obtain some characterizations of KB-spaces.

We introduce a new class of operators that generalizes L-weakly compact operators, which we call order almost L-weakly compact. We give some characterizations of this class and we show that this class of operators satisfies the domination problem.

We characterize Banach lattices $E$ and $F$ on which the adjoint of each operator from $E$ into $F$ which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if $E$ and $F$ are two Banach lattices then each order Dunford-Pettis and weak Dunford-Pettis operator $T$ from $E$ into $F$ has an adjoint Dunford-Pettis operator $T^{\text{'}}$ from $F^{\text{'}}$ into $E^{\text{'}}$ if, and only if, the norm of $E^{\text{'}}$ is order continuous or $F^{\text{'}}$ has the Schur property. As a consequence we show that, if $E$ and $F$ are two Banach...

In this paper, we introduce and study the class of almost weak${}^{☆}$ Dunford-Pettis operators. As consequences, we derive the following interesting results: the domination property of this class of operators and characterizations of the wDP${}^{☆}$ property. Next, we characterize pairs of Banach lattices for which each positive almost weak${}^{☆}$ Dunford-Pettis operator is almost Dunford-Pettis.

We introduce and study the disjoint weak $p$-convergent operators in Banach lattices, and we give a characterization of it in terms of sequences in the positive cones. As an application, we derive the domination and the duality properties of the class of positive disjoint weak $p$-convergent operators. Next, we examine the relationship between disjoint weak $p$-convergent operators and disjoint $p$-convergent operators. Finally, we characterize order bounded disjoint weak $p$-convergent operators in terms...

Let $X$ denote a specific space of the class of $X_{\alpha ,p}$ Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ${\ell }_p$ Banach spaces. We show that for $p>1$ the Banach space $X$ contains asymptotically isometric copies of ${\ell }_p$. It is known that any member of the class is a dual space. We show that the predual of $X$ contains isometric copies of ${\ell }_q$ where $\frac{1}{p}+\frac{1}{q}=1$. For $p=1$ it is known that the predual of the Banach space $X$ contains asymptotically isometric copies of $c_0$. Here we...

We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net $\left(S_{\gamma }\right)$ of compact operators on X such that $su{p}_{\gamma }\left|\right|S_{\gamma }T\left|\right|\le \left|\right|T\left|\right|$ and $S_{\gamma }\to I_X$ in the strong operator topology. Similar results for dual spaces are also proved.

It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^{\xi }$ generated by subsequences $\left(e_{l_n}^{\xi }\right)$ and $\left(e_{m_n}^{\xi }\right)$, respectively, of the natural Schauder basis $\left(e_n^{\xi }\right)$ of $X^{\xi }$ are isomorphic if and only if $\left(e_{l_n}^{\xi }\right)$ and $\left(e_{m_n}^{\xi }\right)$ are equivalent. Further, $X^{\xi }$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $\left(e_n^{\xi }\right)$. It is also shown that there exists a complemented subspace spanned by a block basis of $\left(e_n^{\xi }\right)$, which is not isomorphic to a subspace generated by a subsequence of $\left(e_n^{\zeta }\right)$, for every $0\le \zeta \le \xi$....

We call a subset S of a topological vector space V linearly Borel if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It is shown that a Hamel basis of an infinite-dimensional Banach space can never be linearly Borel. This answers a question of Anatoliĭ Plichko.

These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded operators,...

Denote by F the composition operator generated by a given function f: R --&gt; R, acting on the space of absolutely continuous functions. In this paper we prove that the composition operator F maps the space AC[a,b] into itself if and only if f satisfies a local Lipschitz condition on R.