Weighted Moore-Penrose invertible and weighted EP Banach algebra elements (Q2868511)

The paper under review is concerned with a thorough study of the so-called weighted Moore-Penrose invertible elements and the weighted EP elements of a unital complex Banach algebra \(A\). The case of operators on a Banach space is also considered. The authors call an element \(a\in A\) weighted Moore-Penrose invertible with respect to \(e\) and \(f\) (where \(e\) and \(f\) are positive invertible elements in \(A\)) if there exists (a necessarily unique) \(b\in A\) such that \(a=aba\), \(b=bab\), ab is a hermitian element of the algebra \(A\) equipped with the norm \(\| x\|_e=\| e^{1/2}xe^{-1/2}\|\), and \(ba\) is a hermitian element of \(A\) equipped with the norm \(\| x\|_f=\| f^{1/2}xf^{-1/2}\|\). Further, in the case where \(a\) and \(b\) commute, the authors say that \(a\) is a weighted EP element.