Search

In this paper we study the Y-convexity, a property which is obtained by considering a real Banach sequence lattice Y instead of $\ell^p$ for a linear operator $T : E \rightarrow X$, where E is a Banach space and X is a Banach lattice. We introduce some vector sequence spaces in order to characterize the Y-convexity of T by means of the continuity of an associated operator $\overline{T}$. Analogous results for Y-concavity are also obtained. Finally, the duality between Y-convexity and $Y^{\times}$-concavity is proven., Comment: 31 pages. arXiv admin note: text overlap with arXiv:2311.01124

In this paper, considering a real Banach sequence lattice Y instead of a Lebesgue sequence space $l_p$ we generalize p-convexity of a linear operator $T:E\to X$, where E is a Banach space and X is a Banach lattice. Then we prove that basic properties of p-convexity remain valid for Y-convex linear operators. Analogous generalizations are given for q-concavity and p-summability and composition properties between these operators are analyzed., Comment: 25 pages, currently in peer review phase at Positivity Journal

Given a finite measure space $(\Omega,\Sigma,\mu)$, we show that any Banach space $X(\mu)$ consisting of (equivalence classes of) real measurable functions defined on $\Omega$ such that $f \chi_A \in X(\mu) $ and $ \|f \chi_A \| \leq \|f\|, \, f \in X(\mu), \ A \in \Sigma$, and having the subsequence property, is in fact an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm., Comment: 13 pages