On certain pairs of normal operators (Q2746303)

This note follows a series of papers by the author, in association with others, on certain operator equations. In this paper he gives a new proof of the following theorem.NEWLINENEWLINENEWLINELet \(u\) and \(v\) be commuting invertible normal operators in the W\(^*\)-algebra \(B(H)\) of bounded linear operators on the complex Hilbert space \(H\) and let \(\alpha\) be a complex number not equal to \(\pm 1\), and suppose that \(a\) is an element of \(B(H)\) such that NEWLINE\[NEWLINEuau^{-1}+\alpha u^{-1}au= vav^{-1}+\alpha v^{-1}av.NEWLINE\]NEWLINE Then, NEWLINE\[NEWLINEuau^{-1}= vav^{-1}.NEWLINE\]NEWLINE The author goes on to use his new method of proof to study some properties of the bounded linear mapping \(T\) on \(B(H)\) defined, for \(a\) in \(B(H)\), by NEWLINE\[NEWLINETa= uav.NEWLINE\]