Caractérisations spectrales du radical et du socle d'une paire de Jordan-Banach

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 λ → Sp(f(λ),g(λ))is an analytic multivalued function. From this derives the subharmonicity of the functions λ → ρν(f(lambda;),g(λ)) and λ → log ρν(f(lambda;),g(λ)) where ρ 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.