In this article we deal with the free Banach lattice generated by a and its behavior respect to subspaces. general, any embedding $i\colon \mathbb{L} \longrightarrow \mathbb{M}$ between two lattices $\mathbb{L} \subseteq induces homomorphism $\hat \imath\colon FBL \langle \rangle \mathbb{M}\rangle$ corresponding lattices. We show that mapping \imath$ might not be an isometric neither isomorphic...

We precede the proof with some necessary definitions and notation. Two Banach lattices X and Y are said to be order isometric if there exists an isometry U of X onto Y which preserves the order, that is, U is an isometric surjective operator and U(x) ≥ 0 if and only if x ≥ 0. Let L0 = L0[0, 1] be the vector lattice of all (equivalence classes of) measurable real valued functions on [0, 1] and l...

The Banach Poisson geometry of multi-diagonal Hamiltonian systems having infinitely many integrals in involution is studied. It is shown that these systems can be considered as generalizing the semi-infinite Toda lattice which is an example of a bidiagonal system, a case to which special attention is given. The generic coadjoint orbits of the Banach Lie group of bidiagonal bounded operators are...

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...

It is shown that if (Xn)n is a Bochner integrable stochastic process taking values in a Banach lattice E, the convergence of f(Xn) to f(X) where f is in a total subset of E∗ implies the a.s. convergence. For any Banach space E-valued stochastic process of Pettis integrable strongly measurable functions (Xn)n, the convergence of f(Xn) to f(X) for each f in a total subset of E∗ implies the conver...

We prove complex interpolation theorems for analytic families of multilinear operators defined on quasi-Banach spaces, with explicit constants on the intermediate spaces. We obtain analogous results for analytic families of operators defined on spaces generated by the Calderón method applied to couples of quasi-Banach lattices with nontrivial lattice convexity. As an application we derive a mul...

A net (xα) in a vector lattice X is said to uo-converge to x if |xα−x|∧u o −→ 0 for every u ≥ 0. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several ...