×

On the essential spectra of unbounded operator matrices with non diagonal domain and an application. (English) Zbl 1448.47009

Summary: This paper is devoted to the investigation of the spectral stability of unbounded operator matrices with non diagonal domain in product of Banach spaces. Our results are aimed to characterize some essential spectra of this kind of operators in terms of the union of the essential spectra of the restriction of its diagonal operators entries. The abstract results are illustrated by an example of two-group transport equations with perfect periodic boundary conditions.

MSC:

47A08 Operator matrices
47A53 (Semi-) Fredholm operators; index theories
34K08 Spectral theory of functional-differential operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] F. V. ATKINSON, H. LANGER, R. MENNICKEN ANDA. A. SHKALIKOV, The essential spectrum of some matrix operators, Math. Nachr. 167 (1994), 5-20. · Zbl 0831.47001
[2] A. BATKAI, P. BINDING, A. DIJKSMA, R. HRYNIV,ANDH. LANGER, Spectral problems for operator matrices, Math. Nachr. 278 (2005), 1408-1429. · Zbl 1103.47001
[3] A. BENALI ANDN. MOALLA, Fredholm perturbation theory and some essential spectra in Banach algebra with respect to subalgebra, doi.org/10.1016/j.indag., (2016).
[4] R. DAUTRAY ANDJ. L. LIONS, Analyse math´ematique et calcul num´erique, Masson, Paris 9 (1988).
[5] R. DAUTRAY ANDJ. L. LIONS, Mathematical Analysis and Numerical Methods for Science and Technology, Volume 6: Evolution Problems II, Berlin, Springer, (1999).
[6] I. C. GOHBERG, A. S. MARKUS, I. A. FELDMAN, Normally solvable operators and ideals associated with them, Amer. Math. Soc. Transl., Ser. 2, 61 (1967), 63-84.
[7] S. GOLDBERG, Unbounded operators, New York: Graw-Hill, (1966). · Zbl 0148.12501
[8] G. GREINER, Perturbing the boundary conditions of a generator, Houston J. Math. 13 (1987), 213- 229. · Zbl 0639.47034
[9] A. GROTHENDIECK, Sur les applications lin´eaires faiblement compactes d’´espaces du type C(K), Canad. J. Math. 5 (1953), 129-173. · Zbl 0050.10902
[10] A. JERIBI, Some remarks on the Schechter essential spectra and applications to transport equations, J. Math. Anal. Appl. 275 (2002), 222-237. · Zbl 1045.47063
[11] A. JERIBI, Spectral theory and applications of linear operators and block operator matrices, SpingerVerlag, New-York, (2015). · Zbl 1354.47001
[12] A. JERIBI ANDN. MOALLA, Fredholm operators and Riesz theory for polynomially compact operators, Acta. Appl. Math. 90 (2006), 227-245. · Zbl 1113.47013
[13] A. JERIBI, N. MOALLA ANDI. WALHA, Spectra of some block operator matrices and application to transport operators, J. Math. Anal. Appl. 351 (2009), 315-325. · Zbl 1166.47007
[14] A. JERIBI, N. MOALLA ANDS. YENGUI, S-essential spectra and application to an example of transport operators, Math. Meth. Appl. Sci. doi.org/10.1002/mma.1564, (2012). · Zbl 1300.47013
[15] A. JERIBI, N. MOALLA ANDS. YENGUI, Some results on perturbation theory of matrix operators, M-essential spectra and application to an example of transport operators, http://arxiv.org/licenses/nonexclusive-distrib/1.0/. · Zbl 1300.47013
[16] A. JERIBI ANDI. WALHA, Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectra of some matrix operator and application to a two-group transport equations, Math. Nachr. 284, no. 1, doi.org/10.1002/mana.200710125 (2011), 67-86.
[17] T. KATO, Perturbation theory for nullity deficiency other quantities of linear operators, J. Anal. Math. 6 (1958), 261-322. · Zbl 0090.09003
[18] K. LATRACH, Compactness properties for linear transport operator with abstract bondary conditions with slab geometry, Transport Theory statist. Phys. 22 (1993), 39-65. · Zbl 0774.45006
[19] K. LATRACH ANDA. DEHICI, Fredholm, semi-Fredholm perturbations and essential spectra, J. Math. Anal. Appl. 259 (2001), 227-301. · Zbl 1029.47007
[20] K. LATRACH ANDA. DEHICI, Relatively strictly singular perturbations, essential spectra and application, J. Math. Anal. Appl. 252 (2000), 767-789. · Zbl 0976.47008
[21] V. D. MIL’MAN, Some properties of strictly singular operators, Func. Anal. and Its Appl. 3 (1969), 77-78. · Zbl 0179.17801
[22] N. MOALLA, M. DAMMAK ANDA. JERIBI, Essential spectra of some matrix operators and application to two-group Transport operators with general boundary condition, J. Math. Anal. Appl. 2, 323 (2006), 1071-1090. · Zbl 1108.47014
[23] M. MOKHTAR-KHARROUBI, Time asymptotic behaviour and compactness in neutron transport theory, Euro. Jour. Mech. B Fluid, 11 (1992), 39-68.
[24] V. M ¨ULLER, Spectral theory of linear operator and spectral system in Banach algebras, Oper. Theo. Advances and Appl. 139 (2003).
[25] R. NAGEL, The spectrum of unbounded operator matrices with non-diagonal domain, J. Funct. Anal. 89 (1990), 291-302. · Zbl 0704.47001
[26] A. PELCZYNSKI, On strictly singular and strictly cosingular operators. I. Strictlly singular and cosingular operators on C(Ω) spaces, Bull. Acad. Polo. Sci, S´er. Sci. Math. Astronom. Phys. 13 (1965), 31-36. · Zbl 0138.38604
[27] R. S. PHILLIPS, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516-541. · Zbl 0025.34202
[28] J. QI ANDS. CHEN, Essential spectra of singular matrix differential operators of mixed order, J. Differ. Equ. 250, 13 (2011), 4219-4235. · Zbl 1233.34035
[29] A. A. SHKALIKOV, On the essential spectrum of matrix operators, Math. Notes 58 (1995), 945-949.
[30] L. WEIS, On perturbations of Fredholm operators in Lp(μ)−spaces, Proceeding of the Amer. Math. Soc. 67 (1977), 287-292.
[31] R. J. WHITLEY, Strictly singular operators and their conjugates, Transactions of the Amer. Math. Soc. 113 (1964), 252-261. · Zbl 0124.06603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.