Set approximation by lemniscates and the spectrum of an operator on an interpolation space
James D. Stafney · Pacific Journal of Mathematics · 1975
Let Bo, Bι be an interpolation pair of Banach spaces and T s a bounded linear operator on the corresponding interpolation space [Bo, Bt] s , 0 ^ 5 ^ 1, such that the operators T s all agree on B O ΠB X .In this paper we extend our previous work by giving a general upper bound for the spectrum of T s constructed from the spectra of T o and TΊ 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 T s , O^ks ^ 1, all have the same spectrum.