Displaying similar documents to “On rank one and finite elements of Banach algebras”

Rank, trace and determinant in Banach algebras: generalized Frobenius and Sylvester theorems

Gareth Braatvedt, Rudolf Brits, Francois Schulz (2015)

Studia Mathematica

Similarity:

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.

Finite spectra and quasinilpotent equivalence in Banach algebras

Rudi M. Brits, Heinrich Raubenheimer (2012)

Czechoslovak Mathematical Journal

Similarity:

This paper further investigates the implications of quasinilpotent equivalence for (pairs of) elements belonging to the socle of a semisimple Banach algebra. Specifically, not only does quasinilpotent equivalence of two socle elements imply spectral equality, but also the trace, determinant and spectral multiplicities of the elements must agree. It is hence shown that quasinilpotent equivalence is established by a weaker formula (than that of the spectral semidistance). More generally,...

Rank and the Drazin inverse in Banach algebras

R. M. Brits, L. Lindeboom, H. Raubenheimer (2006)

Studia Mathematica

Similarity:

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.

On rank one elements

Robin Harte (1995)

Studia Mathematica

Similarity:

Without the "scarcity lemma", two kinds of "rank one elements" are identified in semisimple Banach algebras.

Trace and determinant in Banach algebras

Bernard Aupetit, H. Mouton (1996)

Studia Mathematica

Similarity:

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.

Finite rank elements in semisimple Banach algebras

Matej Brešar, Peter Šemrl (1998)

Studia Mathematica

Similarity:

Let A be a semisimple Banach algebra. We define the rank of a nonzero element a in the socle of A to be the minimum of the number of minimal left ideals whose sum contains a. Several characterizations of rank are proved.

Trace and determinant in Jordan-Banach algebras.

Bernard Aupetit, Abdelaziz Maouche (2002)

Publicacions Matemàtiques

Similarity:

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...

Plus-Minus Property as a Generalization of the Daugavet Property

Shepelska, Varvara (2010)

Serdica Mathematical Journal

Similarity:

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.