Closed range multipliers and generalized inverses

Volume 107 / 1993

K. B. Laursen, Studia Mathematica 107 (1993), 127-135 DOI: 10.4064/sm-107-2-127-135

Abstract

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.

Authors

  • K. B. Laursen

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