On complex interpolation and spectral continuity

Saxe, Karen

Saxe, Karen

Studia Mathematica, Tome 129 (1998), p. 223-229 / Harvested from The Polish Digital Mathematics Library

Résumé

Let ${[X_{0}, X_{1}]}_{t}$ , 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both $X_{0}$ and $X_{1}$ will act boundedly on each ${[X_{0}, X_{1}]}_{t}$ . Let $T_{t}$ denote such an operator when considered on ${[X_{0}, X_{1}]}_{t}$ , and $σ (T_{t})$ denote its spectrum. We are motivated by the question of whether or not the map $t \to σ (T_{t})$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: $t \to {(σ (T_{t}))}^{\land}$ (polynomially convex hull) and $t \to \partial_{e} (σ (T_{t}))$ (boundary of the polynomially convex hull). We show that the first of these maps is always upper semicontinuous, and the second is always lower semicontinuous. Using an example from [5], we now have definitive information: $t \to {(σ (T_{t}))}^{\land}$ is upper semicontinuous but not necessarily continuous, and $t \to \partial_{e} (σ (T_{t}))$ is lower semicontinuous but not necessarily continuous.

Publié le : 1998-01-01

Zbl 0915.46015

EUDML-ID : urn:eudml:doc:216554

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     author = {Karen Saxe},
     title = {On complex interpolation and spectral continuity},
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     year = {1998},
     pages = {223-229},
     zbl = {0915.46015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv130i3p223bwm}
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Saxe, Karen. On complex interpolation and spectral continuity. Studia Mathematica, Tome 129 (1998) pp. 223-229. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv130i3p223bwm/

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