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.