On the generalized spectrum of bounded linear operators in Banach spaces

Jue Feng; Xiaoli Li; Kaicheng Fu; Jue Feng; Xiaoli Li; Kaicheng Fu

doi:10.3934/math.2023722

AIMS Mathematics

2023, Volume 8, Issue 6: 14132-14141. doi: 10.3934/math.2023722

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Research article

On the generalized spectrum of bounded linear operators in Banach spaces

School of Mathematical Sciences, Yangzhou University, Yangzhou 225009, Jiangsu, China

Received: 05 February 2023 Revised: 31 March 2023 Accepted: 05 April 2023 Published: 17 April 2023
MSC : 47A10, 47A55

Utilizing the stability characterizations of generalized inverses, we investigate the generalized resolvent of linear operators in Banach spaces. We first prove that the local analyticity of the generalized resolvent is equivalent to the continuity and the local boundedness of generalized inverse functions. We also prove that several properties of the classical spectrum remain true in the case of the generalized one. Finally, we elaborate on the reason why we use the generalized inverse rather than the Moore-Penrose inverse or the group inverse to define the generalized resolvent.
- generalized inverse,
- generalized resolvent,
- generalized spectrum,
- stability characterization
Citation: Jue Feng, Xiaoli Li, Kaicheng Fu. On the generalized spectrum of bounded linear operators in Banach spaces[J]. AIMS Mathematics, 2023, 8(6): 14132-14141. doi: 10.3934/math.2023722

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Abstract

Utilizing the stability characterizations of generalized inverses, we investigate the generalized resolvent of linear operators in Banach spaces. We first prove that the local analyticity of the generalized resolvent is equivalent to the continuity and the local boundedness of generalized inverse functions. We also prove that several properties of the classical spectrum remain true in the case of the generalized one. Finally, we elaborate on the reason why we use the generalized inverse rather than the Moore-Penrose inverse or the group inverse to define the generalized resolvent.

References

[1]	S. J. Chen, Y. Y. Zhao, L. P. Zhu, Q. L. Huang, Regular factorizations and group inverses of linear operators in Banach spaces, Linear Multilinear Algebra, 70 (2022), 1252–1270. http://dx.doi.org/10.1080/03081087.2020.1757604 doi: 10.1080/03081087.2020.1757604
[2]	M. A. Shubin, On holomorphic families of subspaces of a banach space, Integr. Equ. Oper. Theory, 2 (1979), 407–420. http://dx.doi.org/10.1007/BF01682677 doi: 10.1007/BF01682677
[3]	M. Mbekhta, R${\rm\acute{e}}$solvant g$ \rm \acute{e} $n$ \rm \acute{e} $ralis$ \rm \acute{e} $ et th$ \rm \acute{e} $orie spectrale, J. Oper. Theory, 21 (1989), 69–105.
[4]	M. Mbekhta, On the generalized resolvent in Banach spaces, J. Math. Anal. Appl., 189 (1995), 362–377. http://dx.doi.org/10.1006/jmaa.1995.1024 doi: 10.1006/jmaa.1995.1024
[5]	C. Badea, M. Mbekhta, Generalized inverses and the maximal radius of regularity of a Fredholm operator, Integr. Equ. Oper. Theory, 28 (1997), 133–146. http://dx.doi.org/10.1007/BF01191814 doi: 10.1007/BF01191814
[6]	C. Badea, M. Mbekhta, The stability radius of Fredholm linear pencils, J. Math. Anal. Appl., 260 (2001), 159–172. http://dx.doi.org/10.1006/jmaa.2000.7445 doi: 10.1006/jmaa.2000.7445
[7]	A. Hoefer, Reduction of generalized resolvents of linear operator function, Integr. Equ. Oper. Theory, 48 (2004), 479–496. http://dx.doi.org/10.1007/s00020-002-1194-8 doi: 10.1007/s00020-002-1194-8
[8]	Z. Fang, Existence of generalized resolvent of linear bounded operators on Banach space, Nanjing Univ. J. Math. Biquarterly, 22 (2005), 47–52.
[9]	H. F. Ma, H. Hudzik, Y. W. Wang, Z. F. Ma, The generalized regular points and narrow spectrum points of bounded linear operators on Hilbert spaces, Acta Math. Sin.-English Ser., 26 (2010), 2349–2354. http://dx.doi.org/10.1007/s10114-010-9329-3 doi: 10.1007/s10114-010-9329-3
[10]	Q. L. Huang, J. P. Ma, L. Wang, Existence results for generalized resolvents of closed linear operators in Banach spaces, Chin. Ann. Math., 32A (2011), 653–646.
[11]	Q. L. Huang, S. Y. Gao, On the generalized resolvent of linear pencils in Banach spaces, Anal. Theory Appl., 28 (2012), 146–155.
[12]	M. Berkani, S. Č. Živković-Zlatanović, Pseudo-B-Fredholm operators, poles of the resolvent and mean convergence in the Calkin algebra, Filomat, 33 (2019), 3351–3359.
[13]	M. Berkani, On the B-discrete spectrum, Filomat, 34 (2020), 2541–2547.
[14]	J. P. Ma, Complete rank theorem of advanced calculus and singularities of bounded linear operators, Front. Math. China., 3 (2008), 305–316. http://dx.doi.org/10.1007/s11464-008-0019-8 doi: 10.1007/s11464-008-0019-8
[15]	A. Ben-Israel, T. N. E. Greville, Generalized inverses: theory and applications, New York: Springer-Verlag, 2003. https://doi.org/10.1007/b97366

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