Journal of East China Normal University(Natural Science) >

2025 , Vol. 2025 >Issue 1: 13 - 27

DOI: https://doi.org/10.3969/j.issn.1000-5641.2025.01.002

Mathematics

Fa-Weyl’s theorem and a-Weyl’s theorem for bounded linear operators

  • Simeng LI ,
  • Ye ZHANG ,
  • Xiaohong CAO
Expand
  • School of Mathematics and Statistics, Shaanxi Normal University, Xi’an　710119, China

Received date: 2023-11-28

  Online published: 2025-01-20

Copyright

, 2025, Copyright reserved © 2025.
Fold

Abstract

Both Fa-Weyl’s theorem and a-Weyl’s theorem are the variants of Weyl’s theorem. The study of Weyl’s type theorems is very important for spectral theory. By defining a new spectral set in this paper, sufficient and necessary conditions for a bounded linear operator $T $ definded on a Hilbert space to satisfy the Fa-Weyl’s theorem and the a-Weyl’s theorem are established. In addition, we discuss the Fa-Weyl’s theorem and the a-Weyl’s theorem of bounded linear operator $T $ under a finite rank perturbation.

Key words： Fa-Weyl’s theorem; a-Weyl’s theorem; perturbation; spectrum

Cite this article

Simeng LI , Ye ZHANG , Xiaohong CAO . Fa-Weyl’s theorem and a-Weyl’s theorem for bounded linear operators[J]. Journal of East China Normal University(Natural Science), 2025 , 2025(1) : 13 -27 . DOI: 10.3969/j.issn.1000-5641.2025.01.002

References

1 VON WEYL H.. über beschr?nkte quadratische Formen, deren Differenz vollstetig ist. Rendiconti del Circolo Matematico di Palermo, 1909, 27 (1): 373- 392.
2 BERKANI M, KACHAD M.. New Browder and Weyl type theorems. Bulletin of the Korean Mathematical Society, 2015, 52 (2): 439- 452.
3 GUPTA A, MAMTANI K.. Weyl type theorems for unbounded hyponormal operators. Kyungpook Mathematical Journal, 2015, 55 (3): 531- 540.
4 HARTE R, LEE W.. Another note on Weyl’s theorem. Transactions of the American Mathematical Society, 1997, 349 (5): 2115- 2124.
5 DJORDJEVI? D S.. Operators obeying a-Weyl’s theorem. Publicationes Mathematicae Debrecen, 1999, 55 (3/4): 283- 298.
6 BERKANI M, ZARIOUH H.. Generalized a-Weyl’s theorem and perturbations. Functional Analysis, Approximation and Computation, 2010, 2 (1): 7- 18.
7 RASHID M H M.. Property $ (aw) $ and perturbations. Bulletin of the Belgian Mathematical Society-Simon Stevin, 2013, 20 (1): 1- 18.
8 REN Y X, JIANG L N, KONG Y Y.. Property $ (W_{E}) $ and topological uniform descent. Bulletin of the Belgian Mathematical Society-Smion Stevin, 2022, 29 (1): 1- 17.
9 DAI L, CAO X H, GUO Q.. Property $ (\omega) $ and the single-valued extension property. Acta Mathematica Sinica, English Series, 2021, 37 (8): 1254- 1266.
10 SUN C H, CAO X H.. Criteria for the property $ (UWE) $ and the a-Weyl theorem. Functional Analysis and Its Applications, 2022, 56 (3): 216- 224.
11 OUDGHIRI M.. Weyl’s theorem and perturbations. Integral Equations and Operator Theory, 2005, 53 (4): 535- 545.
12 OUDGHIRI M.. A-Weyl’s theorem and perturbations. Studia Mathematica, 2006, 2 (173): 193- 201.
13 AIENA P, TRIOLO S.. Weyl-type theorems on Banach spaces under compact perturbations. Mediterranean Journal of Mathematics, 2018, 15 (3): 1- 18.
14 TAYLOR A E.. Theorems on ascent, descent, nullity and defect of linear operators. Mathematische Annalen, 1966, 163 (1): 18- 49.
15 VLADIMIR M. Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras [M]. Switzerland: Birkh?user Basel, 2003.
16 SCHMOEGER C.. Ein Spektralabbildungssatz. Archiv der Mathematik, 1990, 55, 484- 489.
17 CAO X H, GUO M Z, MENG B.. Weyl spectra and Weyl’s theorem. Journal of Mathematical Analysis and Applications, 2003, 288 (2): 758- 767.
18 AIENA P, BIONDI M T.. Property $ (\omega) $ and perturbations. Journal of Mathematical Analysis and Applications, 2007, 336 (1): 683- 692.
Options
Outlines

/

〈 〉