Vol. 60, No. 2, 1975

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Set approximation by lemniscates and the spectrum of an operator on an interpolation space

James D. Stafney

Vol. 60 (1975), No. 2, 253–265

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.

Mathematical Subject Classification 2000
Primary: 47A10
Secondary: 46M35
Received: 14 January 1974
Published: 1 October 1975
James D. Stafney