Joint spectra and analytic set-valued functions

Abstract:

We investigate analyticity of joint spectra of ${A^m}$-valued holomorphic mappings, where $A$ denotes a complex Banach algebra. We show also that if $K$ is an analytic set-valued function whose values are compact subsets of ${{\mathbf {C}}^n}$ and $d$ is the transfinite diameter in ${{\mathbf {C}}^n}$, then the upper-semicontinuous regularization of $\log d(K)$ is plurisubharmonic. Moreover, we give higher dimensional extensions of Aupetit’s Scarcity Theorem.