# Closed range multipliers and generalized inverses

Studia Mathematica (1993)

- Volume: 107, Issue: 2, page 127-135
- ISSN: 0039-3223

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topLaursen, K., and Mbekhta, M.. "Closed range multipliers and generalized inverses." Studia Mathematica 107.2 (1993): 127-135. <http://eudml.org/doc/216025>.

@article{Laursen1993,

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.},

author = {Laursen, K., Mbekhta, M.},

journal = {Studia Mathematica},

keywords = {closed range of multipliers on general Banach algebras; idempotent; invertible multiplier},

language = {eng},

number = {2},

pages = {127-135},

title = {Closed range multipliers and generalized inverses},

url = {http://eudml.org/doc/216025},

volume = {107},

year = {1993},

}

TY - JOUR

AU - Laursen, K.

AU - Mbekhta, M.

TI - Closed range multipliers and generalized inverses

JO - Studia Mathematica

PY - 1993

VL - 107

IS - 2

SP - 127

EP - 135

AB - 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.

LA - eng

KW - closed range of multipliers on general Banach algebras; idempotent; invertible multiplier

UR - http://eudml.org/doc/216025

ER -

## References

top- P. Aiena and K. B. Laursen, Multipliers with closed range on regular commutative Banach algebras, Proc. Amer. Math. Soc., to appear. Zbl0806.47032
- F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973. Zbl0271.46039
- R. Harte and M. Mbekhta, On generalized inverses in C*-algebras, Studia Math. 103 (1992), 71-77. Zbl0810.46062
- H. Heuser, Functional Analysis, Wiley, New York, 1982. Zbl0465.47001
- B. Host et F. Parreau, Sur un problème de I. Glicksberg: Les idéaux fermés de type fini de M(G), Ann. Inst. Fourier (Grenoble) 28 (3) (1978), 143-164. Zbl0368.43001
- T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261-322. Zbl0090.09003
- R. Larsen, An Introduction to the Theory of Multipliers, Springer, Berlin, 1971. Zbl0213.13301
- C. Rickart, General Theory of Banach Algebras, van Nostrand, Princeton, 1960. Zbl0095.09702

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