Operators in Spaces with Mixed Norm

Abstract

In the present chapter, we study various classes of linear operators acting in spaces with mixed norm and defined in mixed terms of norm and order. If (X, E) is a lattice-normed space and E is a norm lattice of X then X can be endowed with a mixed norm so that X becomes a normed space, and even a Banach space in case the lattice-normed space is br-complete (7.1.1 and 7.1.2). The dual of a space with mixed norm is a space with mixed norm too (7.1.4); furthermore, the canonical embedding into the second dual preserves the vector norm (7.1.5). A more general result states that the space of dominated operators between spaces with mixed norm is itself a space with mixed norm if some natural conditions are met (7.1.9). Passage to the dual of an operator commutes rather often with the taking of the exact dominant of this operator (7.1.10).