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 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 $\mathrm{ran}{Y}^{1/2}\cap \mathrm{ran}{G}^{1/2}=\{0\}$. 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.

References in corpus (1)

• Nonseparability and von Neumann's theorem for domains of unbounded operators