On spectral mapping theorems

Let a bounded domain G in C n 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 C n. 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. Foias. and W. Mlak [Stud. Math. 64 (1979), 263–271], is also discussed.