# Spectral Properties of the Operators *AB* and *BA*

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## Abstract

For linear bounded operators *A, B* from the Banach algebra of linear bounded operators acting in a Banach space, we prove a number of statements on the coincidence of the properties of the operators *I*_{ Y } − *AB*, *I*_{ X } − *BA* related to their kernels and images. In particular, we establish the identical dimension of the kernels, their simultaneous complementability property, the coincidence of the codimensions of the images, their simultaneous Fredholm property and the coincidence of their Fredholm indices. We construct projections onto the image and the kernel of these operators. We establish the simultaneous nonquasianalyticity property of the operators *AB* and *BA*.

## Keywords

linear bounded operator reversibility states spectrum Fredholm property projection operator## Preview

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## References

- 1.N. Bourbaki,
*Théories spectrales*(Hermann, Paris, 1967; Mir, Moscow, 1972).MATHGoogle Scholar - 2.E. B. Davies, “Algebraic aspects of spectral theory,” Mathematica
**57**(1), 63–88 (2011).MathSciNetMATHGoogle Scholar - 3.A. G. Baskakov and K. I. Chernyshov, “Spectral analysis of linear relations and degenerate operator semigroups,” Mat. Sb.
**193**(11), 3–42 (2002) [Sb. Math. 193 (11), 1573–1610 (2002)].MathSciNetCrossRefMATHGoogle Scholar - 4.V. B. Didenko, “On the spectral properties of differential operators with unbounded operator coefficients determined by a linear relation,” Mat. Zametki
**89**(2), 226–240 (2011) [Math. Notes 89 (1–2), 224–237 (2011)].MathSciNetCrossRefMATHGoogle Scholar - 5.V. B. Didenko, “About reversibility states of linear differential operators with periodic unbounded operator coefficients,” Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform.
**14**(2), 136–144 (2014).MATHGoogle Scholar - 6.A. G. Baskakov, “Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations,” Izv. Ross. Akad. Nauk Ser. Mat.
**73**(2), 3–68 (2009) [Izv. Math. 73 (2), 215–278 (2009)].MathSciNetCrossRefMATHGoogle Scholar - 7.A. G. Baskakov and A. Yu. Duplishcheva, “Difference operators and operator-valued matrices of the second order,” Izv. Ross. Akad. Nauk Ser. Mat.
**79**(2), 3–20 (2015) [Izv. Math. 79 (2), 217–232 (2015)].MathSciNetCrossRefGoogle Scholar - 8.A. G. Baskakov, “Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations,” Uspekhi Mat. Nauk
**68**(1 (409)), 77–128 (2013) [Russian Math. Surveys 68 (1), 69–116 (2013)].MathSciNetCrossRefMATHGoogle Scholar - 9.B. A. Barnes, “Common operator properties of the liner operators RS and SR,” Proc. Amer. Math. Soc.
**126**(4), 1055–1061 (1998).MathSciNetCrossRefMATHGoogle Scholar - 10.Yu. I. Lyubich and V. I. Matsaev, “Operators with separable spectrum,” Mat. Sb.
**56**(98) (4), 433–468 (1962).MathSciNetMATHGoogle Scholar - 11.Yu. I. Lyubich, V. I. Matsaev, and G. M. Fel’dman, “On representations with a separable spectrum,” Funktsional. Anal. Prilozhen.
**7**(2), 52–61 (1973) [Functional Anal. Appl. 7 (2), 129–136 (1973)].MathSciNetGoogle Scholar - 12.E. E. Dikarev and D. M. Polyakov, “Harmonic analysis of some classes of linear operators on a real Banach space,” Mat. Zametki
**97**(5), 670–680 (2015) [Math. Notes 97 (5–6), 684–693 (2015)].MathSciNetCrossRefMATHGoogle Scholar - 13.R. Harte, Y. O. Kim, and W. Y. Lee, “Spectral pictures of AB and BA,” Proc. Amer. Math. Soc.
**134**(1), 105–110 (2005).MathSciNetCrossRefMATHGoogle Scholar - 14.Ch. Lin, Z. Yan and Y. Ruan, “Common properties of operators RS and SR and p-hyponormal operators,” Integral Equations Operator Theory
**43**(3), 313–325 (2002).MathSciNetCrossRefMATHGoogle Scholar - 15.Q. P. Zeng and H. J. Zhong, “New results on common properties of bounded linear operators RS and SR,” Acta Math. Sin. (Engl. Ser.)
**29**(10), 1871–1884 (2013).MathSciNetCrossRefMATHGoogle Scholar - 16.A. G. Baskakov and A. I. Pastukhov, “Spectral analysis of a weighted shift operator with unbounded operator coefficients,” Sibirsk. Mat. Zh.
**42**(6), 1231–1243 (2001) [Sib. Math. J. 42 (6), 1026–1036 (2001)].MathSciNetMATHGoogle Scholar - 17.M. S. Bichegkuev, “Linear difference and differential operators with unbounded operator coefficients in weight spaces,” and differential operators with the unbounded operator coefficients in the weighted spaces,” Mat. Zametki
**86**(5–6), 673–680 (2009) [Math. Notes 86 (5–6), 637–644 (2009)].MathSciNetCrossRefMATHGoogle Scholar - 18.M. S. Bichegkuev, “Spectral Analysis of Differential Operators with Unbounded Operator Coefficients in Weighted Spaces of Functions,” Mat. Zametki
**95**(1), 18–25 (2014) [Math. Notes 95 (1–2), 15–21 (2014)].MathSciNetCrossRefMATHGoogle Scholar - 19.A. G. Baskakov and V. B. Didenko, “Spectral analysis of differential operators with unbounded periodic coefficients,” Differ. Uravn.
**51**(3), 323–338 (2015) [Differ. Equations 51 (3), 325–341 (2015)].MathSciNetMATHGoogle Scholar - 20.A. G. Baskakov and V. B. Didenko, “About reversibility states of differential and difference operators,” Izv. Ross. Akad. Nauk Ser. Mat.
**82**(1), 3–16 (2018) [Izv. Math. 82 (1) (2018)].MathSciNetGoogle Scholar

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