Abstract
Penrose showed in [6] that for every complex rectangular matrix a∈Mnxm(ℂ) there exists a unique b∈Mnxm(ℂ), the generalized inverse of a, satisfying the following four conditions: i) a a*b=a, ii) b a*a=a, iii) b b*a=b, iv) a b*b=b.
These condition can be expressed in terms of the triple product <a b c>=a b*c defined in A=Mnxm(ℂ), b* being the adjoint of b. With this triple product and with the operator norm, A is a complex Banach (associative) triple. In this note we will determine all the complex Banach triples with generalized inverses.
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FERNANDEZ LOPEZ, A.,Modular annihilator Jordan algebras. Commun, in Algebra13 (12), 2597–2613 (1985)
FERNANDEZ LOPEZ, A. and GARCIA RUS, E.,Prime associative triple systems with nonzero socle. Submitted to Commun, in Algebra
HESTENES M.R.,A ternary algebra with applications to matrices and linear transformations. Arch. Rational Mech. Anal11, 138–194 (1962)
KAPLANSY, I.,Regular Banach algebras. J. Indian Math. Soc.12, 57–62 (1948)
LOOS, O.,Assoziative tripelsysteme. Manuscripta math.7, 103–112 (1972)
PENROSE, R.,A generalized inverse for matrices. Proc. Cambridge Philos. Soc.51, 406–413 (1955)
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We would like to thank Professor rodriguez Palacios for his interesting suggestions which have enriched our orginal paper.
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López, A.F., Rus, E.G. Banach triples with generalized inverses. Manuscripta Math 62, 503–508 (1988). https://doi.org/10.1007/BF01357724
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DOI: https://doi.org/10.1007/BF01357724