# Closed range multipliers and generalized inverses

Studia Mathematica (1993)

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

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## Abstract

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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.

## How to cite

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Laursen, 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

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1. P. Aiena and K. B. Laursen, Multipliers with closed range on regular commutative Banach algebras, Proc. Amer. Math. Soc., to appear. Zbl0806.47032
2. F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973. Zbl0271.46039
3. R. Harte and M. Mbekhta, On generalized inverses in C*-algebras, Studia Math. 103 (1992), 71-77. Zbl0810.46062
4. H. Heuser, Functional Analysis, Wiley, New York, 1982. Zbl0465.47001
5. 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
6. T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261-322. Zbl0090.09003
7. R. Larsen, An Introduction to the Theory of Multipliers, Springer, Berlin, 1971. Zbl0213.13301
8. C. Rickart, General Theory of Banach Algebras, van Nostrand, Princeton, 1960. Zbl0095.09702

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