# COMMON LOCAL SPECTRAL PROPERTIES OF INTERTWINING LINEAR OPERATORS

Yoo, Jong-Kwang;Han, Hyuk

• Accepted : 2009.05.19
• Published : 2009.06.25
• 26 6

#### Abstract

Let T ${\in}$ $\mathcal{L}$(X), S ${\in}$ $\mathcal{L}$(Y ), A ${\in}$ $\mathcal{L}$(X, Y ) and B ${\in}$ $\mathcal{L}$(Y,X) such that SA = AT, TB = BS, AB = S and BA = T. Then S and T shares that same local spectral properties SVEP, property (${\beta}$), property $({\beta})_{\epsilon}$, property (${\delta}$) and decomposability. From these common local spectral properties, we give some results related with Aluthge transforms and subscalar operators.

#### Keywords

Local Spectral Theory;Bishop's Property ${\beta}$;Decomposable Operators;Subscalar Operators

#### References

1. E. Albrecht, On decomposable operators, Integral Equations Operator Theory, 2(1979), 1-10. https://doi.org/10.1007/BF01729357
2. A. Aluthge, On p-hyponormal operators for 0https://doi.org/10.1007/BF01199886
3. C. Benhida and E. H. Zerouali, Local spectral theory of linear operators RS and SR, Integral Equations Operator Theory, 54(1)(2006), 1-8. https://doi.org/10.1007/s00020-005-1375-3
4. Lin Chen, Yan Zikun and Ruan Yingbin, Common operator properties of operators RS and SR and p-Hyponormal operators, Integral Equations Operator Theory, 43(2002), 313-325. https://doi.org/10.1007/BF01255566
5. I. Colojoarva and C. Foias, Theory of Generalized Spectral Operators, Gorden and Breach, New York, 1968.
6. J. Eschmeier and M. Putinar, Bishop's condition ($\beta$) and rich extensions of linear operators, Indiana Univ. Math. J., 37(1988), 325-348. https://doi.org/10.1512/iumj.1988.37.37016
7. 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.
8. B. Malgrange, Ideals of Differentiable Functions, Oxford University Press, London, 1967.
9. M. Putinar, Hyponormal operators are subscalar, J. Operator Theory, 12(1984), 385-395.