Abstract
We obtain a description of all pairs of Hermitian operators X and Y, which satisfy the condition −Y ≤ X ≤ Y. We give the examples of such operator pairs. Each of the presented examples leads us to the new weak majorization for the Hermitian operator pair. It is shown that this inequality does not necessarily imply the inequality |X| ≤ ZY Z* for any operator Z, ‖Z‖ ≤ 1. We prove that invertibility of Y follows from invertibility of operators X and A* A for Hermitian operators X and Y, Y ≥ 0 and an arbitrary operator A such that −AY A* ≤ X ≤ AY A*. We discuss one analog of triangle inequality found by the author in one earlier paper for pairs of Hermitian operators.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.