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.

No abstract

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. p(x) = 0 ⇔ x = 0; p(λx) = |λ|p(x) for all λ ∈ R, x ∈ X; and p(x + y) ≤ p(x) + p(y) for all x, y ∈ X) then the triple (X, p, E) is called a lattice-normed space, abbreviated as LN S; see for example [15]. If X is a vector lattice and the vector norm p is monotone (i.e. |x| ≤ |y| ⇒ p(x) ≤ p(y)) then the triple (X, p, E) is called a lattice-normed vector lattice,