Spectrum of interpolated operators

Albrecht, Ernst; Müller, Vladimir

doi:10.1090/S0002-9939-00-05862-7

Spectrum of interpolated operators
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by Ernst Albrecht and Vladimir Müller PDF

Proc. Amer. Math. Soc. 129 (2001), 807-814 Request permission

Abstract:

Let $(X_0,X_1)$ be a compatible pair of Banach spaces and let $T$ be an operator that acts boundedly on both $X_0$ and $X_1$. Let $T_{[\theta ]} \quad (0\le \theta \le 1)$ be the corresponding operator on the complex interpolation space $(X_0,X_1)_{[\theta ]}$. The aim of this paper is to study the spectral properties of $T_{[\theta ]}$. We show that in general the set-valued function $\theta \mapsto \sigma (T_{[\theta ]})$ is discontinuous even in inner points $\theta \in (0,1)$ and show that each operator satisfies the local uniqueness-of-resolvent condition of Ransford. Further we study connections with the real interpolation method.

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