Bull. Korean Math. Soc. 2011; 48(6): 1129-1135
Printed November 1, 2011
https://doi.org/10.4134/BKMS.2011.48.6.1129
Copyright © The Korean Mathematical Society.
Jong-Kwang Yoo
Chodang University
In this note, we prove that every subscalar operator with finite spectrum is algebraic. In particular, a quasi-nilpotent subscalar operator is nilpotent. We also prove that every subscalar operator with property $(\delta)$ on a Banach space of dimension greater than 1 has a non-trivial invariant closed linear subspace.
Keywords: algebraic spectral subspace, analytic spectral subspace, decomposable operator, invariant subspaces property $(\beta)$, property $(\delta),$ subscalar operator
MSC numbers: Primary 47A11, 47A53
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