Laursen, K. B.; Mbekhta, Mostafa Closed range multipliers and generalized inverse. (English) Zbl 0812.47031 Stud. Math. 107, No. 2, 127-135 (1993). Summary: Conditions involving closed range of multipliers on general Banach algebras are studied. Numerous conditions equivalent to a splitting \(A= TA\oplus \ker T\) are listed for a multiplier \(T\) defined on the Banach algebra \(A\). For instance, it is shown that \(TA\oplus \ker T= A\) if and only if there is a commuting operator \(S\) for which \(T= TST\) and \(S= STS\), that this is the case if and only if such \(S\) may be taken to be a multiplier, and that these conditions are also equivalent to the existence of a factorization \(T= PB\), where \(P\) is an idempotent and \(B\) an invertible multiplier. The latter condition establishes a connection to a famous problem of harmonic analysis. Cited in 13 Documents MSC: 47B48 Linear operators on Banach algebras 46H05 General theory of topological algebras 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 46L05 General theory of \(C^*\)-algebras Keywords:closed range of multipliers on general Banach algebras; idempotent; invertible multiplier PDFBibTeX XMLCite \textit{K. B. Laursen} and \textit{M. Mbekhta}, Stud. Math. 107, No. 2, 127--135 (1993; Zbl 0812.47031) Full Text: EuDML