# WEYL'S TYPE THEOREMS FOR ALGEBRAICALLY (p, k)-QUASIHYPONORMAL OPERATORS

• Published : 2012.01.31

#### Abstract

For a bounded linear operator T we prove the following assertions: (a) If T is algebraically (p, k)-quasihyponormal, then T is a-isoloid, polaroid, reguloid and a-polaroid. (b) If $T^*$ is algebraically (p, k)-quasihyponormal, then a-Weyl's theorem holds for f(T) for every $f{\in}Hol({\sigma}T))$, where $Hol({\sigma}(T))$ is the space of all functions that analytic in an open neighborhoods of ${\sigma}(T)$ of T. (c) If $T^*$ is algebraically (p, k)-quasihyponormal, then generalized a-Weyl's theorem holds for f(T) for every $f{\in}Hol({\sigma}T))$. (d) If T is a (p, k)-quasihyponormal operator, then the spectral mapping theorem holds for semi-B-essential approximate point spectrum $\sigma_{SBF_+^-}(T)$, and for left Drazin spectrum ${\sigma}_{lD}(T)$ for every $f{\in}Hol({\sigma}T))$.

#### References

1. P. Aiena, Fredholm and Local Spectral Theory with Applications to Multipliers, Kluwer, 2004.
2. P. Aiena and O. Monsalve, The single valued extension property and the generalized Kato decomposition property, Acta Sci. Math. (Szeged) 67 (2001), no. 3-4, 791-807.
3. M. Amouch and H. Zguitti, On the equivalence of Browder's and generalized Browder's theorem, Glasg. Math. J. 48 (2006), no. 1, 179-185. https://doi.org/10.1017/S0017089505002971
4. S. K. Berberian, An extension of Weyl's theorem to a class of not necessarily normal operators, Michigan Math. J. 16 (1969), 273-279. https://doi.org/10.1307/mmj/1029000272
5. S. K. Berberian, The Weyl spectrum of an operator, Indiana Univ. Math. J. 20 (1970), 529-544. https://doi.org/10.1512/iumj.1970.20.20044
6. M. Berkani, On a class of Quasi-Fredholm operators, Integral Equations Operator Theory 34 (1999), no. 2, 244-249. https://doi.org/10.1007/BF01236475
7. M. Berkani, Index of B-Fredholm operators and generalization of a Weyl theorem, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1717-1723. https://doi.org/10.1090/S0002-9939-01-06291-8
8. M. Berkani, B-Weyl spectrum and poles of the resolvent, J. Math. Anal. Appl. 272 (2002), no. 2, 596-603. https://doi.org/10.1016/S0022-247X(02)00179-8
9. M. Berkani, On the equivalence of Weyl theorem and generalized Weyl theorem, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 1, 103-110. https://doi.org/10.1007/s10114-005-0720-4
10. M. Berkani and A. Arroud, Generalized weyl's theorem and hyponormal operators, J. Aust. Math. Soc. 76 (2004), no. 2, 1-12. https://doi.org/10.1017/S1446788700008661
11. M. Berkani and J. Koliha, Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), no. 1-2, 359-376.
12. M. Berkani and M. Sarih, An Atkinson-type theorem for B-Fredholm operators, Studia Math. 148 (2001), no. 3, 251-257. https://doi.org/10.4064/sm148-3-4
13. L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285-288. https://doi.org/10.1307/mmj/1031732778
14. R. E. Curto and Y. M. Han, Weyl's theorem for algebraically paranormal operators, Integral Equations Operator Theory 47 (2003), no. 3, 307-314. https://doi.org/10.1007/s00020-002-1164-1
15. H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, London, 1978.
16. B. P. Duggal and S. V. Djordjevic, Generalized Weyl's theorem for a class of operators satisfying a norm condition, Proc. Roy. Irish Acad. Sect. A 104A (2004), no. 1, 271-277.
17. B. P. Duggal and S. V. Djordjevic, Generalised Weyl's theorem for a class of operators satisfying a norm condition, Math. Proc. R. Ir. Acad. 104A (2004), no. 1, 75-81.
18. J. K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58 (1975), no. 1, 61-69. https://doi.org/10.2140/pjm.1975.58.61
19. R. E. Harte, Invertibility and Singularity for Bounded Linear Operators, Marcel Dekker, New York, 1988.
20. S. H. Lee and W. Y. Lee, On Weyl's theorem II, Math. Japon. 43 (1996), no. 3, 549-553.
21. A.-H. Kim and S. U. Yoo, Weyl's theorem for isoloid and reguloid operators, Commun. Korean Math. Soc. 14 (1999), no. 1, 179-188.
22. I. H. Kim, On (p, k)-quasihyponormal operators, Math. Inequal. Appl. 7 (2004), no. 4, 629-638.
23. J. J. Koliha, Isolated spectral points, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3417-3424. https://doi.org/10.1090/S0002-9939-96-03449-1
24. M. Lahrouz and M. Zohry, Weyl type theorems and the approximate point spectrum, Irish Math. Soc. Bull. 55 (2005), 41-51.
25. K. B. Laursen, Operators with finite ascent, Pacific J. Math. 152 (1992), no. 2, 323-336. https://doi.org/10.2140/pjm.1992.152.323
26. S. Mecheri, Weyl's theorem for algebraically (p, k)-quasihyponormal operators, Georgian Math. J. 13 (2006), no. 2, 307-313.
27. V. Rakocevic, Operators obeying a-Weyl's theorem, Rev. Roumaine Math. Pures Appl. 34 (1989), no. 10, 915-919.
28. V. Rakocevic, Semi-Fredholm operators with finite ascent or descent and perturbation, Proc. Amer. Math. Soc. 122 (1995), no. 12, 3823-3825.
29. V. Rakocevic, Semi-Browder operators and perturbations, Studia Math. 122 (1997), no. 2, 131-137. https://doi.org/10.4064/sm-122-2-131-137
30. V. Rakocevic, Operators Obeying a-Weyl's theorem, Publ. Math. Debrecen 55 (1999), no. 3-4, 283-298.
31. M. H. M. Rashid, M. S. M. Noorani, and A. S. Saari, Weyl's type theorems for quasi-Class A operators, J. Math. Stat. 4 (2008), no. 2, 70-74. https://doi.org/10.3844/jmssp.2008.70.74
32. M. H. M. Rashid, M. S. M. Noorani, and A. S. Saari, Generalized Weyl's theorem for log-hyponormal, Malaysian J. Math. Soc. 2 (2008), no. 1, 73-82.
33. C. Schmoeger, On operators T such that Weyl's theorem holds for f(T), Extracta Math. 13 (1998), no. 1, 27-33.
34. K. Tanahashi, A. Uchiyama, and M. Cho, Isolated point of spectrum of (p, k)-quasihyponormal operators, Linear Algebra Appl. 382 (2004), no. 1, 221-229. https://doi.org/10.1016/j.laa.2003.12.021
35. A. E. Taylor, Theorems on ascent, descent, nullity and defect of linear operators, Math. Ann. 163 (1966), 18-49. https://doi.org/10.1007/BF02052483
36. H. Weyl, Uber beschrankte quadratische Formen, deren Differenze vollsteting ist, Rend. Circ. Math. Palermo 27 (1909), 373-392. https://doi.org/10.1007/BF03019655
37. H. Zguitti, A note on generalized Weyl's theorem, J. Math. Anal. Appl. 316 (2006), no. 1, 373-381. https://doi.org/10.1016/j.jmaa.2005.04.057

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