Geodesic
Spectral Properties of the Operators~$AB$ and $BA$
Matematičeskie zametki, Tome 103 (2018) no. 2, pp. 196-209.

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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. 
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D. B. Didenko. Spectral Properties of the Operators~$AB$ and $BA$. Matematičeskie zametki, Tome 103 (2018) no. 2, pp. 196-209. http://geodesic.mathdoc.fr/item/MZM_2018_103_2_a3/

[1] N. Burbaki, Spektralnaya teoriya, Mir, M., 1972 | MR | Zbl

[2] E. B. Davies, “Algebraic aspects of spectral theory”, Mathematica, 57:1 (2011), 63–88 | MR | Zbl

[3] A. G. Baskakov, K. I. Chernyshov, “Spektralnyi analiz lineinykh otnoshenii i vyrozhdennye polugruppy operatorov”, Matem. sb., 193:11 (2002), 3–42 | DOI | MR | Zbl

[4] V. B. Didenko, “O spektralnykh svoistvakh differentsialnykh operatorov s neogranichennymi operatornymi koeffitsientami, opredelyaemykh lineinym otnosheniem”, Matem. zametki, 89:2 (2011), 226–240 | DOI | MR | Zbl

[5] V. B. Didenko, “O sostoyaniyakh obratimosti lineinykh differentsialnykh operatorov s neogranichennymi periodicheskimi koeffitsientami”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:2 (2014), 136–144 | Zbl

[6] A. G. Baskakov, “Spektralnyi analiz differentsialnykh operatorov s neogranichennymi operatornymi koeffitsientami, raznostnye otnosheniya i polugruppy raznostnykh otnoshenii”, Izv. RAN. Ser. matem., 73:2 (2009), 3–68 | DOI | MR | Zbl

[7] A. G. Baskakov, A. Yu. Duplischeva, “Raznostnye operatory i operatornye matritsy vtorogo poryadka”, Izv. RAN. Ser. matem., 79:2 (2015), 3–20 | DOI | MR | Zbl

[8] A. G. Baskakov, “Issledovanie lineinykh differentsialnykh uravnenii metodami spektralnoi teorii raznostnykh operatorov i lineinykh otnoshenii”, UMN, 68:1 (409) (2013), 77–128 | DOI | MR | Zbl

[9] B. A. Barnes, “Common operator properties of the liner operators $RS$ and $SR$”, Proc. Amer. Math. Soc., 126:4 (1998), 1055–1061 | DOI | MR | Zbl

[10] Yu. I. Lyubich, V. I. Matsaev, “Ob operatorakh s otdelimym spektrom”, Matem. sb., 56 (98):4 (1962), 433–468 | MR | Zbl

[11] Yu. I. Lyubich, V. I. Matsaev, G. M. Feldman, “O predstavleniyakh s otdelimym spektrom”, Funkts. analiz i ego pril., 7:2 (1973), 52–61 | MR | Zbl

[12] E. E. Dikarev, D. M. Polyakov, “Garmonicheskii analiz nekotorykh klassov lineinykh operatorov v veschestvennom banakhovom prostranstve”, Matem. zametki, 97:5 (2015), 670–680 | DOI | MR | Zbl

[13] R. Harte, Y. O. Kim, W. Y. Lee, “Spectral pictures of $AB$ and $BA$”, Proc. Amer. Math. Soc., 134:1 (2005), 105–110 | DOI | MR

[14] Ch.Lin, Z. Yan, Y. Ruan, “Common properties of operators $RS$ and $SR$ and $p$-hyponormal operators”, Integral Equations Operator Theory, 43:3 (2002), 313–325 | DOI | MR | Zbl

[15] Q. P. Zeng, H. J. Zhong, “New results on common properties of bounded linear operators $RS$ and $SR$”, Acta Math. Sin. (Engl. Ser.), 29:10 (2013), 1871–1884 | DOI | MR | Zbl

[16] A. G. Baskakov, A. I. Pastukhov, “Spektralnyi analiz operatora vzveshennogo sdviga s neogranichennymi operatornymi koeffitsientami”, Sib. matem. zhurn., 42:6 (2001), 1231–1243 | MR | Zbl

[17] M. S. Bichegkuev, “Lineinye raznostnye i differentsialnye operatory s neogranichennymi operatornymi koeffitsientami v vesovykh prostranstvakh”, Matem. zametki, 86:5 (2009), 673–680 | DOI | MR | Zbl

[18] M. S. Bichegkuev, “Spektralnyi analiz differentsialnykh operatorov s neogranichennymi operatornymi koeffitsientami v vesovykh prostranstvakh funktsii”, Matem. zametki, 95:1 (2014), 18–25 | DOI | MR | Zbl

[19] A. G. Baskakov, V. B. Didenko, “Spektralnyi analiz differentsialnykh operatorov s neogranichennymi periodicheskimi koeffitsientami”, Differents. uravneniya, 51:3 (2015), 323–338 | DOI | MR | Zbl

[20] A. G. Baskakov, V. B. Didenko, “O sostoyaniyakh obratimosti raznostnykh i differentsialnykh operatorov”, Izv. RAN. Ser. matem., 82:1 (2018), 3–16