全部 标题 作者
关键词 摘要


Quasinilpotent Part of w-Hyponormal Operators

DOI: 10.4236/oalib.1100548, PP. 1-15

Subject Areas: Functional Analysis, Mathematical Analysis

Keywords: Aluthge Transformation, w-Hyponormal Operators, Polaroid Operators, Reguloid Operators, SVEP, Property β, Quasinilpotent Part

Full-Text   Cite this paper   Add to My Lib

Abstract

For a w-hyponormal operator T acting on a separable complex Hilbert space H, we prove that: 1) the quasi-nilpotent part Ho(- λ) is equal to Ker(T- λI); 2) has Bishop’s property<i>β</i>; 3) if σw (T)={0}, then it is a compact normal operator; 4) If T is an algebraically w-hyponormal operator, then it is polaroid and reguloid. Among other things, we prove that ifTn and Tn* are w-hyponormal, then T is normal.

Cite this paper

Rashid, M. (2014). Quasinilpotent Part of w-Hyponormal Operators. Open Access Library Journal, 1, e548. doi: http://dx.doi.org/10.4236/oalib.1100548.

References

[1]  Aluthge, A. (1990) On p-Hyponormal Operators for 0<p≤1. Integral Equation Operator Theory, 13, 307-315.
http://dx.doi.org/10.1007/BF01199886
[2]  Ito, M. (1999) Some Classes of Operators Associated with Generalized Aluthge Transformation. SUT Journal of Mathematics, 35, 149-165.
[3]  Aluthge, A. and Wang, D. (2000) w-Hyponormal Operators. Integral Equation Operator Theory, 36, 1-10.
http://dx.doi.org/10.1007/BF01236285
[4]  Uchiyama, A. (1999) Berger-Shaw’s Theorem for p-Hyponormal Operators. Integral Equations and Operator Theory, 33, 221-230.
http://dx.doi.org/10.1007/BF01233965
[5]  Uchiyama, A. (1999) Inequalities of Putnam and Berger-Shaw for p-Quasihyponormal Operators. Integral Equations and Operator Theory, 34, 91-106.
http://dx.doi.org/10.1007/BF01332494
[6]  Ito, M. and Yamazaki, T. (2002) Relations between Two Equalities and and Their Applications. Integral Equation Operator Theory, 44, 442-450.
http://dx.doi.org/10.1007/BF01193670
[7]  Aluthge, A. and Wang, D. (2000) w-Hyponormal Operators II. Integral Equation Operator Theory, 37, 324-331.
http://dx.doi.org/10.1007/BF01194481
[8]  Yanagida, M. (2002) Powers of Class wA(s,t)Operators Associated with Generalized Aluthge Transformation. Journal of Inequalities and Applications, 7, 143-168.
[9]  Finch, J.K. (1975) The Single Valued Extension Property on a Banach Space. Pacific Journal of Mathematics, 58, 61-69.
http://dx.doi.org/10.2140/pjm.1975.58.61
[10]  Laursen, K.B. (1992) Operators with Finite Ascent. Pacific Journal of Mathematics, 152, 323-336.
http://dx.doi.org/10.2140/pjm.1992.152.323
[11]  Bishop, E. (1959) A Duality Theorem for an Arbitrary Operator. Pacific Journal of Mathematics, 9, 379-397.
http://dx.doi.org/10.2140/pjm.1959.9.379
[12]  Kimura, F. (1995) Analysis of Non-Normal Operators via Aluthge Transformation. Integral Equations and Operator Theory, 50, 375-384.
http://dx.doi.org/10.1007/s00020-003-1231-2
[13]  Duggal, B.P. (2001) p-Hyponormal Operators Satisfy Bishop’s Condition (β). Integral Equations and Operator Theory, 40, 436-440.
http://dx.doi.org/10.1007/BF01198138
[14]  Aiena, P. (2004) Fredholm and Local Spectral Theory with Applications to Multipliers. Kluwer.
[15]  Laursen, K.B. and Neumann, M.M. (2000) An Introduction to Local Spectral Theory. Clarendon, Oxford.
[16]  Coburn, L.A. (1966) Weyl’s Theorem for Nonnormal Operators. Michigan Mathematical Journal, 13, 285-288.
http://dx.doi.org/10.1307/mmj/1031732778
[17]  Han, Y.M., Lee, J.I. and Wang, D. (2005) Riesz Idempotent and Weyl’s Theorem for w-Hyponormal Operators. Integral Equations and Operator Theory, 53, 51-60.
http://dx.doi.org/10.1007/s00020-003-1313-1
[18]  Harte, R.E. (1988) Invertibility and Singularity for Bounded Linear Operators. Dekker, New York.
[19]  Berkani, M. and Sarih, M. (2001) An Atkinson-Type Theorem for B-Fredholm Opera-tors. Studia Mathematica, 148, 251-257.
http://dx.doi.org/10.4064/sm148-3-4
[20]  Koliha, J.J. (1996) Isolated Spectral Points. Proceedings of the American Mathematical Society, 124, 3417-3424.
http://dx.doi.org/10.1090/S0002-9939-96-03449-1
[21]  Berkani, M. and Koliha, J. (2003) Weyl Type Theorems for Bounded Linear Operators. Acta Scientiarum Mathematicarum (Szeged), 69, 359-376.
[22]  Berkani, M. (2007) On the Equivalence of Weyl Theorem and Generalized Weyl Theorem. Acta Mathematica Sinica, 23, 103-110.
http://dx.doi.org/10.1007/s10114-005-0720-4
[23]  Rakocevic, V. (1989) Operators Obeying α-Weyl’s Theorem. Revue Roumaine de Mathématiques Pures et Appliquées, 34, 915-919.
[24]  Aiena, P. and Pena, P. (2006) Variations on Weyl’s Theorem. Journal of Mathematical Analysis and Applications, 324, 566-579.
http://dx.doi.org/10.1016/j.jmaa.2005.11.027
[25]  Aiena, P. and Miller, T.L. (2007) On Generalized α-Browder’s Theorem. Studia Mathematica, 180, 285-300.
http://dx.doi.org/10.4064/sm180-3-7
[26]  Amouch, M. and Berkani, M. (2008) On the Property (gw). Mediterranean Journal of Mathematics, 5, 371-378.
http://dx.doi.org/10.1007/s00009-008-0156-z
[27]  Amouch, M. and Zguitti, H. (2006) On the Equivalence of Browder’s and Generalized Browder’s Theorem. Glasgow Mathematical Journal, 48, 179-185.
http://dx.doi.org/10.1017/S0017089505002971
[28]  Aiena, P., Guillen, J. and Pena, P. (2008) Property (w) for Perturbations of Polaroid Operators. Linear Algebra and Its Applications, 428, 1791-1802.
http://dx.doi.org/10.1016/j.laa.2007.10.022
[29]  Stampfli, J.G. (1962) Hyponormal Operators. Pacific Journal of Mathematics, 12, 1453-1458.
http://dx.doi.org/10.2140/pjm.1962.12.1453

Full-Text


comments powered by Disqus