### Consistent invertibility and the judgement of Weyls theorem

LIU Ying, CAO Xiao-hong*

1. School of Mathematics and Information Science, Shaanxi Normal University, Xian 710062, Shaanxi, China
• Published:2019-06-05

Abstract: H is an infinite dimensional separable complex Hilbert space and B(H) is the algebra of all bounded linear operators on H. An operator T∈B(H) is said to be “consistent in invertibility” provided that for each S∈B(H), TS and ST are both or neither invertible. Based on the property of consistency in invertibility, we give the necessary and sufficient conditions for T and its functional calculus which the Weyls theorem hold.

CLC Number:

• O177.2
 [1] KATO T. Perturbation theory for linear operator[M]. New York: Springer-Verlag, 1966.[2] GOHBERG I, GOLDBERG S, KAASHORK M A. Unbounded linear operators[J]. American Mathematical Monthly, 1968, 75(7):288-322.[3] HARTE R. On Kato non-singularity[J]. Studia Mathematica,1996, 117(2):107-114.[4] GONG Weibang, HAN Deguang. Spectrum of the products of operators and compact perturbations[J]. Proceedings of The American Mathematical Society, 1994, 120(3):755-760.[5] DJORDJEVIC D S. Operators consistent in regularity[J]. Publicationes Mathematicae Debrecen, 2002, 60(1):1-15.[6] WEYL H V. Über beschränkte quadratische formen, deren differenz vollstetig ist[J]. Rendiconti Del Circolo Matematico Di Palermo, 1909, 27(1):373-392.[7] BERBERIAN S K. An extension of Weyls theorem to a class of not necessarily normal operators[J]. Michigan Mathematical Journal, 1969, 16(3):273-279.[8] LI Chunguang, ZHU Sen, FENG Youling. Weyls theorem for functions of operators and approximation[J]. Integral Equations and Operator Theory, 2010, 67(4):481-497.[9] CURTO R E, HAN Y M. Weyls theorem for algebraically paranormal operators[J]. Integral Equations and Operator Theory, 2003, 47(3):307-341.[10] AN I J, HAN Y M. Weyls theorem for algebraically quasi-class a operators[J]. Integral Equations and Operator Theory, 2008, 62(1):1-10.[11] SHI Weijuan, CAO Xiaohong. Weyls theorem for the square of operator and perturbations[J]. Communications in Contemporary Mathematics, 2015, 17(1):1-11.[12] COBURN L A. Weyls theorem for nonnormal operators[J]. Michigan Mathematical Journal, 1966, 13(3):285-288.[13] DUGGAL B P. The Weyl spectrum of p-hyponormal operators[J]. Integral Equations and Operator Theory, 1997, 29(2):197-201.[14] CAO Xiaohong. Analytically class operators and Weyls theorem[J]. Journal of Mathematical Analysis and Applications, 2006, 320(2):795-803.[15] TAYLOR A E. Theorems on ascent, descent, nullity and defect of linear operators[J]. Mathematische Annalen, 1996, 163(1):18-49.[16] HARTE R. Invertibility and singularity for bounded linear operators[M]. New York: Marcel Dekker, 1988.[17] HARTE R, LEE W Y. Another note on Weyls theorem[J]. Transactions of the American Mathematical Society, 1997, 349(5):2115-2124.
 [1] ZHOU An-min, HU Lei, LIU Lu-ping, JIA Peng, LIU Liang. Malicious Office document detection technology based on entropy time series [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(5): 1-7. [2] Zhao-xia WU,Jia-qi WANG. Wireless single spectrum secure auction algorithm [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(11): 51-55. [3] ZHANG Ying, CAO Xiao-hong, DAI Lei. Judgement of Weyls theorem for bounded linear operators [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 82-87. [4] SONG Jia-jia, CAO Xiao-hong, DAI Lei. The judgement for the small compact perturbation of SVEP for upper triangular operator matrices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(4): 61-67. [5] DAI Lei, CAO Xiao-hong. Property(z)and Weyl type theorem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(2): 60-65. [6] KONG Ying-ying, CAO Xiao-hong, DAI Lei. Judgement of a-Weyls theorem and its perturbations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 77-83. [7] WU Xue-li, CAO Xiao-hong, ZHANG Min. The perturbation of the single valued extension property for bounded linear operators [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(12): 5-9. [8] YANG Gong-lin, JI Pei-sheng. Some properties of primitive ideal submodules in Hilbert C*-modules [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(10): 50-55. [9] CHEN Shi-zhao, CAO Xiao-hong*. Linear maps between operator algebras preserving the ascent and descent [J]. J4, 2013, 48(12): 86-89. [10] LU Shi-fang1, WEI Liang2, ZHAO Hai-xing2. On signless Laplace integral graphs of complete tripartite graphs [J]. J4, 2012, 47(12): 41-46. [11] GAO Jie. Structure of eigenvalues of multi-point boundary value problems [J]. J4, 2011, 46(8): 17-22. [12] ZHANG He-jia, CAO Xiao-hong*. The equivalence of a-Browder theorem and property (ω1) for operational calculus of operators [J]. J4, 2011, 46(4): 108-112. [13] WANG Ji-rong1, CAO Xiao-hong2, LIU Jun-ying2. Operators with consistency in Fredholm and Weyl′s theorem [J]. J4, 2011, 46(1): 87-91. [14] WANG Ji-rong1, CAO Xiao-hong2. On the perturbation of the Kato essential spectra for upper  triangular operator matrices [J]. J4, 2010, 45(3): 90-95. [15] ZHAO Ling-ling, ZHANG He-jia, CAO Xiao-hong*. Essential spectrum of the products of operators [J]. J4, 2010, 45(10): 83-88.
Viewed
Full text

Abstract

Cited

Shared
Discussed