Geodesic
Closed range multipliers and generalized inverses
Studia Mathematica, Tome 107 (1993) no. 2, pp. 127-135

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DOI

Conditions involving closed range of multipliers on general Banach algebras are studied. Numerous conditions equivalent to a splitting A = TA ⊕ kerT are listed, for a multiplier T defined on the Banach algebra A. For instance, it is shown that TA ⊕ kerT = 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. 
K. B. Laursen;  . Closed range multipliers and generalized inverses. Studia Mathematica, Tome 107 (1993) no. 2, pp. 127-135. doi: 10.4064/sm-107-2-127-135
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