Finite Operators and Weyl Type Theorems for Quasi-*-n-Paranormal Operators

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 4,  2015, pp.885-892
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.4.885
Title & Authors
Finite Operators and Weyl Type Theorems for Quasi-*-n-Paranormal Operators
ZUO, FEI; YAN, WEI;

Abstract
In this paper, we mainly obtain the following assertions: (1) If T is a quasi-*-n-paranormal operator, then T is finite and simply polaroid. (2) If T or $\small{T^*}$ is a quasi-*-n-paranormal operator, then Weyl's theorem holds for f(T), where f is an analytic function on $\small{{\sigma}(T)}$ and is not constant on each connected component of the open set U containing $\small{{\sigma}(T)}$. (3) If E is the Riesz idempotent for a nonzero isolated point $\small{{\lambda}}$ of the spectrum of a quasi-*-n-paranormal operator, then E is self-adjoint and $\small{EH=N(T-{\lambda})=N(T-{\lambda})^*}$.
Keywords
Quasi-*-n-paranormal operator;finite;polaroid;Weyl's theorem;Riesz idempotent;
Language
English
Cited by
References
1.
P. Aiena, Fredholm and Local Spectral Theory with Applications to Multipliers, Kluwer Academic Publishers, London, 2004.

2.
P. Aiena, E. Aponte and E. Balzan, Weyl type theorems for left and right polaroid operators, Integr. Equ. Oper. Theory, 66(1)(2010), 1-20.

3.
T. Furuta, Invitation to Linear Operators, Taylor & Francis, London, 2001.

4.
I. H. Jeon and I. H. Kim, On operators satisfying $T^*|T^2|T{\geq}T^*|T|^2T$, Linear Algebra Appl., 418(2006), 854-862.

5.
K. B. Laursen and M. M. Neumann, Introduction to Local Spectral Theory, Clarendon Press, Oxford, 2000.

6.
M. Y. Lee, S. H. Lee and C. S. Rhoo, Some remarks on the structure of k-$\ast$-paranormal operators, Kyungpook Math. J., 35(1995), 205-211.

7.
S. Mecheri, Finite operators, Demonstratio Math., 35(2)(2002), 357-366.

8.
M. Oudghiri, Weyl's theorem and purturbations, Integr. Equ. Oper. Theory, 53(4)(2005), 535-545.

9.
S. M. Patel, Contributions to the Study of Spectraloid Operators, PhD, Delhi Univ, DE, India, 1974.

10.
V. Rakocevic, Operators obeying a-Weyl's theorem, Rev. Roumaine Math. Pures Appl., 34(10)(1989), 915-919.

11.
J. L. Shen and Alatancang, The spectral properties of quasi-$\ast$-paranormal operators, Chinese Annals of Mathematics (China), 34(6)(2013), 663-670.

12.
J. Stampfli, Hyponormal operators and spectral density, Trans. Amer. Math. Soc., 117(1965), 469-476.

13.
K. Tanahashi, I. H. Jeon, I. H. Kim and A. Uchiyama, Quasinilpotent part of class A or (p, k)-quasihyponormal operators, Operator Theory, Advances and Applications, 187(2008), 199-210.

14.
K. Tanahashi and A. Uchiyama, A note on $\ast$-paranormal operators and related classes of operators, Bull. Korean Math. Soc., 51(2)(2014), 357-371.

15.
A. Uchiyama, On the isolated points of the spectrum of paranomal operators, Integr. Equ. Oper. Theory, 55(2006), 145-151.

16.
J. P. Williams, Finite operators, Proc. Amer. Math. Soc., 26(1970), 129-135.

17.
J. T. Yuan and Z. S. Gao, Weyl spectrum of class A(n) and n-paranomal operators, Integr. Equ. Oper. Theory, 60(2008), 289-298.

18.
F. Zuo, On quasi-$\ast$-n-paranormal operators, J. Math. Inequal., 9(2)(2015), 409-415.

19.
F. Zuo and J. L. Shen, A note on $\ast$-n-paranormal operators, Adv. Math. (China), 42(2)(2013), 156-163.