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A CHARACTERIZATION OF LOCAL RESOLVENT SETS

  • Han Hyuk (Department of Mathematics Seonam University) ;
  • Yoo Jong-Kwang (Department of Liberal Arts and Science Chodang University)
  • Published : 2006.04.01

Abstract

Let T be a bounded linear operator on a Banach space X. And let ${{\rho}T}(X)$ be the local resolvent set of T at $x\;{\in}\;X$. Then we prove that a complex number ${\lambda}$ belongs to ${{\rho}T}(X)$ if and only if there is a sequence $\{x_{n}\}$ in X such that $x_n\;=\;(T - {\lambda})x_{n+1}$ for n = 0, 1, 2,..., $x_0$ = x and $\{{\parallel}x_n{\parallel}^{\frac{1}{n}}\}$ is bounded.

Keywords

local spectral theory

References

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