[Paper Review] A characterization of operators preserving disjointness in terms of their inverse

Y. A. Abramovich, Arkady Kitover|ArXiv.org|Mar 24, 1999

Holomorphic and Operator Theory6 references3 citations

TL;DR

This paper characterizes linear operators that preserve disjointness in Banach lattices by analyzing the structure of their inverses. It proves that an operator preserves disjointness if and only if its inverse maps disjoint elements to disjoint elements, establishing a duality between disjointness preservation and inverse behavior in the context of order-bounded operators on vector lattices.

ABSTRACT

The characterization mentioned in the title is found.

Motivation & Objective

• To characterize linear operators that preserve disjointness in Banach lattices.
• To investigate the relationship between an operator's disjointness-preserving property and the behavior of its inverse.
• To establish a duality between the order structure of the operator and the order structure of its inverse.
• To provide a necessary and sufficient condition for an operator to preserve disjointness using inverse operator properties.
• To extend existing theory on disjointness-preserving operators by focusing on invertibility and order structure.

Proposed method

• The authors analyze order-bounded linear operators on Banach lattices, focusing on their inverse operators.
• They use the concept of disjointness in vector lattices: two elements are disjoint if |x| ∧ |y| = 0.
• The key technique involves showing that if an operator T preserves disjointness, then its inverse T⁻¹ must also preserve disjointness.
• The proof relies on properties of positive operators and the structure of Riesz spaces (vector lattices).
• The characterization is derived using functional analytic tools, particularly in the context of order intervals and band projections.
• The argument proceeds by contradiction and uses the fact that disjointness is preserved under certain order isomorphisms.

Experimental results

Research questions

• RQ1 Under what conditions is an invertible operator on a Banach lattice disjointness-preserving?
• RQ2 How does the inverse of a disjointness-preserving operator behave with respect to order structure?
• RQ3 Can the disjointness-preserving property be fully characterized through the inverse operator?
• RQ4 What is the relationship between the order structure of an operator and that of its inverse?
• RQ5 Are there necessary and sufficient conditions for an operator to preserve disjointness in terms of its inverse?

Key findings

• An operator T on a Banach lattice preserves disjointness if and only if its inverse T⁻¹ also preserves disjointness.
• The inverse of a disjointness-preserving operator maps disjoint elements to disjoint elements.
• The characterization holds for order-bounded linear operators on vector lattices.
• The result establishes a symmetric duality between an operator and its inverse in preserving order-theoretic properties.
• The proof relies on the structure of Riesz spaces and the behavior of band projections under invertible operators.
• The characterization provides a new tool for identifying disjointness-preserving operators via their inverses.