Heinrich Raubenheimer (2010)
Czechoslovak Mathematical Journal
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We characterize elements in a semisimple Banach algebra which are quasinilpotent equivalent to maximal finite rank elements.
Gareth Braatvedt, Rudolf Brits, Francois Schulz (2015)
Studia Mathematica
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As a follow-up to a paper of Aupetit and Mouton (1996), we consider the spectral definitions of rank, trace and determinant applied to elements in a general Banach algebra. We prove a generalization of Sylvester's Determinant Theorem to Banach algebras and thereafter a generalization of the Frobenius inequality.
T. Mouton, H. Raubenheimer (1993)
Studia Mathematica
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We give a spectral characterisation of rank one elements and of the socle of a semisimple Banach algebra.
R. M. Brits, L. Lindeboom, H. Raubenheimer (2006)
Studia Mathematica
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Let A be an arbitrary, unital and semisimple Banach algebra with nonzero socle. We investigate the relationship between the spectral rank (defined by B. Aupetit and H. Mouton) and the Drazin index for elements belonging to the socle of A. In particular, we show that the results for the finite-dimensional case can be extended to the (infinite-dimensional) socle of A.
Teresa Álvarez (2014)
Colloquium Mathematicae
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We describe the Browder Riesz-Schauder theory of compact operators in Banach spaces in the context of polynomially finite rank linear relations in Banach spaces.
Bernard Aupetit, H. Mouton (1996)
Studia Mathematica
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We show that the trace and the determinant on a semisimple Banach algebra can be defined in a purely spectral and analytic way and then we obtain many consequences from these new definitions.
Bernard Aupetit, Abdelaziz Maouche (2002)
Publicacions Matemàtiques
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Using an appropriate definition of the multiplicity of a spectral value, we introduce a new definition of the trace and determinant of elements with finite spectrum in Jordan-Banach algebras. We first extend a result obtained by J. Zemánek in the associative case, on the connectedness of projections which are close to each other spectrally (Theorem 2.3). Secondly we show that the rank of the Riesz projection associated to a finite-rank element a and a finite subset of its spectrum is...
Jaroslav Zemánek (1982)
Banach Center Publications
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Bernard Aupetit (1982)
Banach Center Publications
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Shepelska, Varvara (2010)
Serdica Mathematical Journal
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2000 Mathematics Subject Classification: Primary 46B20. Secondary 47A99, 46B42. It was shown in [2] that the most natural equalities valid for every rank-one operator T in real Banach spaces lead either to the Daugavet equation ||I+T|| = 1 + ||T|| or to the equation ||I − T|| = ||I+T||. We study if the spaces where the latter condition is satisfied for every finite-rank operator inherit the properties of Daugavet spaces.