On spectral mapping theorems

AbstractLet a bounded domain G in Cn be either strictly pseudoconvex with C2-boundary b(G) or a polydomain. For each u = (u1,…, uk) ⊂ H∞(G), u ⊂ A(G), resp. let \̃gs(u) be a compact subset of Cn. If so defined \̃gs is a subspectrum [i.e., satisfies the projection property and \̃gs(u) is a non-void subset of the “usual” joint spectrum of u], then it is shown that u(\̃gs(z)∩ G) ⊂ \̃gs(u). Moreover, if u is continuously extendable to each point of \̃gs(z) ∩ b(G), then u(\̃gs(z)) = \̃gs(u). This provides spectral mapping theorems for H∞(G) [resp.A(G)]-functional calculi. The extended spectrum of a representation, introduced by C. Foiaş. and W. Mlak [Stud. Math. 64 (1979), 263–271], is also discussed