The parallel sum for adjointable operators on Hilbert $C^{\ast }$-modules

Abstract

The parallel sum for adjoinable operators on Hilbert $C^{\ast }$-modules is introduced and studied. Some results known for matrices and bounded linear operators on Hilbert spaces are generalized to the case of adjointable operators on Hilbert $C^{\ast }$-modules. It is shown that there exist a Hilbert $C^{\ast }$-module $H$ and two positive operators $A,B\in \mathcal{L}(H)$ such that the operator equation $A^{1/2}=(A+B{)}^{1/2}X,X\in \mathcal{L}(\mathcal{H})$ has no solution, where $\mathcal{L}(H)$ denotes the set of all adjointable operators on $H$.

20 pages. Accepted for publication in ACTA Mathematica Sinica (Chinese Series). Proposition 4.6 is improved