Compact-like operators in lattice-normed spaces

Abstract

A linear operator T between two vector lattices normed by locally solid Riesz spaces is said to be p τ -continuous if, for any p τ -null net (x α ), the net (T x α ) is p τ -null, and T is said to be p τ -bounded operator if it sends p τ -bounded subsets to p τ -bounded subsets. Also, T is called p τ -compact if, for any p τ -bounded net (x α ), the net (T x α ) has a p τ -convergent subnet. They generalize several known classes of operators such as norm continuous, order continuous, p-continuous, order bounded, p-bounded, compact and AM-compact operators. We study the general properties of these operators.

Nonstandard hulls of lattice-normed ordered vector spaces

“…We define and study certain necessary notions such as p -normality and op -continuity in LNOVSs, p-Levi spaces, etc. (see also [4][5][6] for their lattice versions).…”

“…We continue with further basic notions in LNOVSs, which are motivated by their analogies for vector lattices and for LNVLs (see, for example, [4][5][6]19,20,[25][26][27]). …”

“…Note that a vector lattice E can be seen as the corresponding LNS (E, | · |, E). Lattice-normed vector lattices (abbreviated by LNVLs) have attracted attention in [4,5,10,12,14,26,27]. The general theory of lattice-normed ordered vector spaces (abbreviated by LNOVSs) is still under investigation.…”

“…The structure of the paper is as follows. In Section 2, we include elementary theory of LNOVSs in parallel with theory of LNVLs (see, for example, [4,5,26]). In Section 3, we introduce and investigate the nonstandard hull of an LNS normed by a Dedekind complete vector lattice.…”

Unbounded p-Convergence in Lattice-Normed Vector Lattices

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Let (x α ) be a net in a vector lattice normed by locally solid lattice (X, p, E τ ). We say that (This convergence has been studied recently for lattice-normed vector lattices as the up-convergence in [5,6,7], the uo-convergence in [14], and, as the un-convergence in [10,13,14,16,18]. In this paper, we study the general properties of the unboundedLet X be a vector space, E be a vector lattice, and p : X → E + be a vector norm (i.e.

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