A CHARACTERIZATION OF LOCAL RESOLVENT SETS

Title & Authors
A CHARACTERIZATION OF LOCAL RESOLVENT SETS
Han Hyuk; Yoo Jong-Kwang;

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