On the mappings connected with parallel addition of nonnegative operators

TL;DR

This paper investigates a specific operator mapping related to parallel addition of nonnegative operators, providing explicit formulas, analyzing its properties, and exploring its connections to operator decompositions and domain intersections.

Contribution

It introduces and characterizes the mapping u_G, deriving explicit expressions and analyzing its properties and applications in operator theory.

Findings

u_G is sub-additive and homogeneous of degree one.

The image of u_G consists of operators with disjoint square root ranges.

Connections to Lebesgue-type decompositions and domain intersection properties are established.

Abstract

We study a mapping $\tau_G$ of the cone ${\mathbf B}^+({\mathcal H})$ of bounded nonnegative self-adjoint operators in a complex Hilbert space ${\mathcal H}$ into itself. This mapping is defined as a strong limit of iterates of the mapping ${\mathbf B}^+({\mathcal H})\ni X\mapsto\mu_G(X)=X-X:G\in{\mathbf B}^+({\mathcal H})$, where $G\in{\mathbf B}^+({\mathcal H})$ and $X:G$ is the parallel sum. We find explicit expressions for $\tau_G$ and establish its properties. In particular, it is shown that $\tau_G$ is sub-additive, homogeneous of degree one, and its image coincides with set of its fixed points which is the subset of ${\mathbf B}^+({\mathcal H})$, consisting of all $Y$ such that ${\rm ran\,} Y^{1/2}\cap{\rm ran\,} G^{1/2}=\{0\}$. Relationships between $\tau_G$ and Lebesgue type decomposition of nonnegative self-adjoint operator are established and applications to the properties of unbounded self-adjoint operators with trivial intersections of their domains are given.