New results on common properties of bounded linear operators \(RS\) and \(SR\) (Q381057)

Let \(X,Y\) be Banach spaces, \(R:X\rightarrow Y\) and \(S:Y\rightarrow X\) be bounded linear operators. In the present paper, the authors introduce new regularities (in the sense of [\textit{V. Kordula} and \textit{V. Müller}, Stud. Math. 119, No.~2, 109--128 (1996; Zbl 0857.47001)]) and show that the equality \(\sigma_{*}(RS)\backslash\{0\}=\sigma_{*}(SR)\backslash\{0\}\) holds for the spectra corresponding to these new regularities. Among other results, the authors show that, for all nonzero \(\lambda\in\mathbb{C}\), \(\lambda I-SR\) has complemented kernel if and only if \(\lambda I-RS\) has complemented kernel. At the end, some applications to \(B\)-Fredholm theory, extension and Aluthge transforms are given.