Lattice properties of strength functions

Abstract

This paper investigates an important functional representation of the cone of bounded positive semidefinite operators. It is known that the representation by strength functions turns the Löwner order into the pointwise order. However, very little is known about the structure of strength functions. Our main result says that the representation behaves naturally with the infimum and supremum operations. More precisely, we show that the pointwise minimum of two strength functions \(f_A\) and \(f_B\) is a strength function if and only if the infimum of A and B exists. This complements a recent result of L. Molnár stating that the pointwise maximum of \(f_A\) and \(f_B\) exists if and only if A and B are comparable, as this latter statement is equivalent to the existence of the supremum. The cornerstone of each argument in this paper is a fact that was discovered recently, namely that the strength function of the parallel sum A : B (which is half of the harmonic mean) equals the parallel sum of the strength functions \(f_A\) and \(f_B\). We provide a new proof for this statement, and as a byproduct, in some special cases, we describe the strength function of the so-called (generalized) short.