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# Caractérisations spectrales du radical et du socle d'une paire de jordan-banach

Published:1997-12-01
Printed: Dec 1997
• Abdelaziz Maouche
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## Abstract

If $f$ and $g$ are two analytic functions from a domain $D$ of the complex plane into respectively the Banach spaces $V^+$ and $V^-$, we prove that $\lambda\mapsto \Sp\bigl(f(\lambda),g(\lambda)\bigr)$ is an analytic multivalued function. From this derives the subharmonicity of the functions $\lambda\mapsto \rho_V\bigl(f(\lambda),g(\lambda)\bigr)$ and $\lambda\mapsto \log\rho_V\bigl(f(\lambda),g(\lambda)\bigr)$ where $\rho$ denotes the spectral radius. We apply these results to obtain nice caracterizations of the radical and the socle of a Banach Jordan pair, and finally we get an algebraic structural theorem.
 Keywords: Spectre, rayon spectral, multifonction analytique, quasi-inverse, paire de Jordan-Banach, radical de Jacobson, socle.
 MSC Classifications: 46H70 - Nonassociative topological algebras [See also 46K70, 46L70] (17A15) - unknown classification (17A15)