# 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
• 64 6

#### 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

1. I. Colojoara and C. Foias, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968
2. I. Erdelyi and R. Lange, Spectral Decompositions on Banach Spaces, Lecture Notes in Math. 623, Springer Verlag, New York, 1977
3. K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, London Mathematical Society Monographs New Series 20, Oxford Science Publications, Oxford, 2000
4. M. Martin and M. Putinar, Lectures on Hyponormal Operators, Birkhauser Verlag, Basel Boston Berlin, 1989
5. D. Xia, Spectral Theory of Hyponormal Operators, Birkhauser Verlag, Basel Boston Berlin, 1983
6. K. B. Laursen and P. Vrbova, Some remarks on the surjectivity spectrum of linear operators, Czechoslovak Math. J. 39(114) (1989), 730-739