On the mappings connected with parallel addition of nonnegative operators

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 describe its properties. In particular, it is shown that $\tau _G$ is sub-additive, homogeneous of degree one, and its image coincides with the set of its fixed points which is a subset of ${\mathbf B}^+({\mathcal H})$, consisting of all Y such that $\mathrm{ran\,} Y^{\frac{1}{2}}\cap \mathrm{ran\,} G^{\frac{1}{2}}=\{0\}$. 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.