CLOSED RANGE MULTIPLIERS AND GENERALIZED INVERSES

Conditions involving closed range of multipliers on general Banach alg ebras are studied. Numerous conditions equivalent to a splitting A = T A + ker T axe listed, for a multiplier T defined on the Banach algebra A. For instance, it is shown that TA + 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 t hat these conditions are also equivalent to the existence of a factori zation 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.