Annals of Functional Analysis

Common properties of the operator products in spectral theory

Xiaochun Fang and Kai Yan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $X, Y$ be Banach spaces, $A, D: X\rightarrow Y$ and $B, C: Y\rightarrow X$ be the bounded linear operators satisfying operator equation set $$\left\{ \begin{aligned} ACD=DBD~ \\ DBA=ACA. \\ \end{aligned} \right.. $$ The concept of regularity was firstly introduced by Kordula and M$\ddot{u}$ller. In this paper, we investigate the common properties of $AC$ and $BD$ in viewpoint of regularity when $A, B, C$ and $D$ all satisfy the operator equation set above.

Article information

Source
Ann. Funct. Anal., Volume 6, Number 4 (2015), 60-69.

Dates
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.afa/1435764001

Digital Object Identifier
doi:10.15352/afa/06-4-60

Mathematical Reviews number (MathSciNet)
MR3365981

Zentralblatt MATH identifier
1334.47003

Subjects
Primary: 47A10: Spectrum, resolvent
Secondary: 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]

Keywords
Jacobson's lemma operator equation set common property regularity

Citation

Yan, Kai; Fang, Xiaochun. Common properties of the operator products in spectral theory. Ann. Funct. Anal. 6 (2015), no. 4, 60--69. doi:10.15352/afa/06-4-60. https://projecteuclid.org/euclid.afa/1435764001


Export citation

References

  • B.A. Barnes, Common operator properties of the linear operators $RS$ and $SR$, Proc. Amer. Math. Soc. 126 (1998), no. 4. 1055–1061.
  • C. Benhida and E.H. Zerouali, Local Spectral Theory of Linear Operators $RS$ and $SR$, Integral Equations Operator Theory 54 (2006), no. 1, 1–8.
  • M. Berkani, Restriction of an operator to the range of its powers, Studia Math. 140 (2000), no. 2, 163-175.
  • G. Corach, B. Duggal and R. Harte, Extensions of Jacobson's lemma, Comm. Algebra 41 (2013), no. 2, 520-531.
  • S. Grabiner, Uniform ascent and descent of bounded operators, J. Math. Soc. Japan 34 (1982), no. 2, 317-337.
  • R. Harte, Spectral Mapping Theorems A Bluffer's Guide, Springer, 2014.
  • M.A. Kaashoek, Ascent, descent, nullity and defect, a note on a paper by A. E. Taylor, Math. Ann. 172 (1967), no. 2, 105-115.
  • V. Kordula and V. M$\ddot{u}$ller, On the axiomatic theory of spectrum, Studia Math. 119 (1996), no. 2. 109-128.
  • C. Lin, Z. Yan and Y. Ruan, Common properties of operators $RS$ and $SR$ and $p$-hyponormal operators, Integral Equations Operator Theory 43 (2002), no. 3. 313-325.
  • M. Mbekhta and V. Müller, On the axiomatic theory II, Studia Math. 119 (1996), no. 2. 129-147.
  • V. M$\ddot{\makebox{u}}$ller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, second edition, Birkh$\ddot{\makebox{a}}$user, Basel, Boston, Berlin, 2007.
  • K. Yan and X.C. Fang, Common properties of the operator products in local spectral theory, submitted.
  • Q.P. Zeng and H.J. Zhong, Common properties of bounded linear operators $AC$ and $BA$: Spectral theory, Math. Nachr. 267 (2014), no. 5-6. 717-725.
  • Q.P. Zeng and H.J. Zhong, New results on common properties of the bounded linear operators $RS$ and $SR$, Acta Math. Sinica (English Series) 29 (2013), no. 10. 1871-1884.