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

http://dx.doi.org/10.4153/CMB-1997-058-9
Canad. Math. Bull. 40(1997), 488-497

  Published:1997-12-01
 Printed: Dec 1997

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.