We complete a result of Hernandez on the complex interpolation for families of Banach lattices.

Equivalent formulations of the Dunford-Pettis property of order $p$ (${DPP}_p$), $1<p<\infty$, are studied. Let $L(X,Y)$, $W(X,Y)$, $K(X,Y)$, $U(X,Y)$, and $C_p(X,Y)$ denote respectively the sets of all bounded linear, weakly compact, compact, unconditionally converging, and $p$-convergent operators from $X$ to $Y$. Classical results of Kalton are used to study the complementability of the spaces $W(X,Y)$ and $K(X,Y)$ in the space $C_p(X,Y)$, and of $C_p(X,Y)$ in $U(X,Y)$ and $L(X,Y)$.

Let X,Y,A and B be Banach spaces such that X is isomorphic to Y ⊕ A and Y is isomorphic to X ⊕ B. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In the present paper, we give a necessary and sufficient condition on sextuples (p,q,r,s,u,v) in ℕ with p + q ≥ 2, r + s ≥ 1 and u, v ∈ ℕ* for X to be isomorphic to Y whenever these spaces satisfy the following decomposition scheme: ⎧ $X^u\sim X^p\oplus Y^q$, ⎨ ⎩ $Y^v\sim A^r\oplus B^s$. Namely, Ω = (p-u)(s-r-v)...

For a fusion Banach frame $(\{G_n,v_n\},S)$ for a Banach space $E$, if $(\{v_n^{*}\left(E^{*}\right),v_n^{*}\},T)$ is a fusion Banach frame for $E^{*}$, then $(\{G_n,v_n\},S;\{v_n^{*}\left(E^{*}\right),v_n^{*}\},T)$ is called a fusion bi-Banach frame for $E$. It is proved that if $E$ has an atomic decomposition, then $E$ also has a fusion bi-Banach frame. Also, a sufficient condition for the existence of a fusion bi-Banach frame is given. Finally, a characterization of fusion bi-Banach frames is given.

It is shown that every strongly lattice norm on $c_0\left(\Gamma \right)$ can be approximated by $C^{\infty }$ smooth norms. We also show that there is no lattice and Gâteaux differentiable norm on $C_0[0,{\omega }_1]$.

The present paper is devoted to some applications of the notion of L-Dunford-Pettis sets to several classes of operators on Banach lattices. More precisely, we establish some characterizations of weak Dunford-Pettis, Dunford-Pettis completely continuous, and weak almost Dunford-Pettis operators. Next, we study the relationships between L-Dunford-Pettis, and Dunford-Pettis (relatively compact) sets in topological dual Banach spaces.

We show that the following well-known open problems on existence of Lipschitz isomorphisms between subsets of Hilbert spaces are equivalent: Are balls isomorphic to spheres? Is the whole space isomorphic to the half space?