## Annals of Functional Analysis

### Common properties of the operator products in spectral theory

#### 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

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

#### 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

#### 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.