On matching distance between eigenvalues of unbounded operators

Yıl 2022, Cilt: 5 Sayı: 1, 46 - 53, 14.03.2022

Micheal Gil

https://doi.org/10.33205/cma.1060718

Öz

Let $A$ and $\tilde{A}$ be linear operators on a Banach space having compact resolvents, and let ${\lambda }_k\left(A\right)$ and ${\lambda }_k\left(\tilde{A}\right)(k=1,2,\dots )$ be the eigenvalues taken with their algebraic multiplicities of $A$ and $\tilde{A}$, respectively. Under some conditions, we derive a bound for the quantity
$$\mathrm{md}(A,\tilde{A}):=\underset{\pi }{inf}\underset{k=1,2,\dots }{sup}\left|{\lambda }_{\pi \left(k\right)}\right(\tilde{A})-{\lambda }_k(A\left)\right|$$
where $\pi$ is taken over all permutations of the set of all positive integers. That quantity is called the matching optimal distance between the eigenvalues of $A$ and $\tilde{A}$. Applications of the obtained bound to matrix differential operators are also discussed.

Anahtar Kelimeler

Banach space, perturbations of eigenvalues, matching distance, differential operator, tensor product of Hilbert spaces

Kaynakça

• B. Abdelmoumen, A. Jeribi and M. Mnif: Invariance of the Schechter essential spectrum under polynomially compact operator perturbation, Extracta Math., 26 (1) (2011), 61–73.
• P. Aiena, S. Triolo: Some perturbation results through localized SVEP, Acta Sci. Math. (Szeged), 82 (1–2) (2016), 205–219.
• A. D. Baranov, D. V. Yakubovich: Completeness of rank one perturbations of normal operators with lacunary spectrum, J. Spectr. Theory, 8 (1) (2018), 1–32.
• S. Buterin, S.V. Vasiliev: On uniqueness of recovering the convolution integro-differential operator from the spectrum of its non-smooth one-dimensional perturbation, Bound. Value Probl., (2018), Paper No. 55, 12 pp.
• W. Chaker, A. Jeribi and B. Krichen: Demicompact linear operators, essential spectrum and some perturbation results, Math. Nachr., 288 (13) (2015), 1476–1486.
• N. Dunford, J.T. Schwartz: Linear Operators, part I. General Theory,Wiley Interscience publishers, New York (1966).
• M. I. Gil: Perturbations of operators on tensor products and spectrum localization of matrix differential operators, J. Appl. Funct. Anal., 3 (3) (2008), 315–332.
• M. I. Gil: Spectral approximations of unbounded non-selfadjoint operators, Analysis and Mathem. Physics, 3 (1) (2013), 37–44.
• M. I. Gil: Spectral approximations of unbounded operators of the type "normal plus compact", Funct. Approximatio. Comment. Math., 51 (1) (2014), 133–140.
• M. I. Gil: Operator Functions and Operator Equations.World Scientific, New Jersey (2018).
• M. I. Gil: Norm estimates for resolvents of linear operators in a Banach space and spectral variations, Advances in Operator Theory, 4 (1) (2019), 113–139.
• A. Jeribi: Spectral Theory and Applications of Linear Operators and Block Operator Matrices, Springer-Verlag, New-York (2015).
• A. Jeribi: Linear Operators and Their Essential Pseudospectra, CRC Press, Boca Raton (2018).
• A. Jeribi: Perturbation Theory for Linear Operators: Denseness and Bases with Applications, Springer-Verlag, Singapore (2021).
• T. Kato: Perturbation Theory for Linear Operators, Springer-Verlag, Berlin (1980).
• R. Killip: Perturbations of one-dimensional Schrodinger operators preserving the absolutely continuous spectrum, Int. Math. Res. Not., 38 (2002), 2029-2061.
• R. Ma, H. Wang and M. Elsanosi: Spectrum of a linear fourth-order differential operator and its applications, Math. Nachr., 286 (17-18) (2013), 1805-1819.
• M. L. Sahari, A. K. Taha and L. Randriamihamison: A note on the spectrum of diagonal perturbation of weighted shift operator, Matematiche (Catania), 74 (1) (2019), 35-47.
• G. W. Stewart, Ji-guang Sun: Matrix Perturbation Theory, Academic Press, New York (1990).
• M. Zhang, J. Sun and J. Ao: The discreteness of spectrum for higher-order differential operators in weighted function spaces. Bull. Aust. Math. Soc., 86 (3) (2012), 370-376.