The parallel sum for adjointable operators on Hilbert $C^*$-modules

TL;DR

This paper introduces and explores the concept of the parallel sum for adjointable operators on Hilbert $C^*$-modules, extending known results from matrices and Hilbert space operators to this broader setting.

Contribution

It generalizes the parallel sum concept to Hilbert $C^*$-modules and demonstrates the existence of operators where certain operator equations have no solutions.

Findings

Generalization of parallel sum to Hilbert $C^*$-modules

Existence of operators with no solution to a specific operator equation

Extension of known matrix/operator results to $C^*$-modules

Abstract

The parallel sum for adjoinable operators on Hilbert $C^*$-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 $C^*$-modules. It is shown that there exist a Hilbert $C^*$-module $H$ and two positive operators $A, B\in\mathcal{L}(H)$ such that the operator equation $A^{1/2}=(A+B)^{1/2}X, X\in \cal{L}(H)$ has no solution, where $\mathcal{L}(H)$ denotes the set of all adjointable operators on $H$.