A functional representation approach to vector lattice covers for spaces of compact operators

TL;DR

The paper develops an order-theoretic framework for spaces of bounded linear operators between ordered normed spaces, focusing on when the positive operator cone $L(X,Y)_+$ has non-empty interior and on representing the closure $C(X,Y)$ of finite-rank operators as a vector lattice cover. It proves a general functional representation result that embeds such spaces into spaces of continuous functions on compact sets determined by extremal functionals, and uses this to characterize extremal functionals on $C(X,Y)'_+$ and to derive practical criteria for disjointness and modulus. A key contribution is a precise interiority criterion for $L(X,Y)_+$ and a constructive vector lattice cover for $C(X,Y)$ under broad conditions, enabling explicit tests for disjointness and modulus and providing concrete examples. The results connect finite-dimensional polyhedral cases with infinite-dimensional settings (e.g., $\ell^1$ and Loewner cones on Hilbert spaces) and offer a unified toolkit to analyze operator spaces via function-space representations. This framework enhances our ability to transfer order-structure questions about operator spaces to functional-analytic questions on compact spaces, with potential applications in analysis and mathematical physics.

Abstract

For ordered normed vector spaces $X, Y$, we consider the space $\mathcal{L}(X,Y)$ of bounded linear operators and characterize when its cone of positive operators has non-empty interior. When this is satisfied, we give a functional representation of the closure $\mathcal{C}(X,Y)$ of the finite rank operators in $\mathcal{L}(X,Y)$. This space is particularly interesting since it coincides in many cases with the space of compact operators from $X$ to $Y$. Our functional representation has very good order properties in the sense that it is a so-called vector lattice cover of $\mathcal{C}(X,Y)$. This can be used to characterize disjointness of operators in $\mathcal{C}(X,Y)$ and to determine which operators have a modulus in $\mathcal{C}(X,Y)$. We demonstrate how our results can be applied to a variety of concrete spaces.

Theorems & Definitions (29)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• Proposition 2.5
• Theorem 3.1
• proof
• Corollary 3.2
• proof