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Closed range multipliers and generalized inverse. (English) Zbl 0812.47031

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.

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
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