Let B0, B1 be an interpolation
pair of Banach spaces and Ts a bounded linear operator on the corresponding
interpolation space [B0,B1]s,0 ≦ s ≦ 1, such that the operators Ts all agree on
B0∩B1. In this paper we extend our previous work by giving a general upper bound
for the spectrum of Ts constructed from the spectra of T0 and T1 using a set
interpolation formula which we introduce in §1 for compact sets in the plane. In §3
we show that this upper bound is essentially best possible. This requires a theorem
about approximating sets with lemniscates, which we prove in §2. Finally, we show in
§4 that under certain conditions the operators Ts,0 ≦ s ≦ 1, all have the same
spectrum.
