Stability of the essential spectrum of the diagonally and off-diagonally dominant block matrix linear relations

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• 作者：Aymen Ammar ; Slim Fakhfakh ; Aref Jeribi
• 关键词：Matrix of linear relations ; Diagonally dominant operators
• 刊名：Journal of Pseudo-Differential Operators and Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：7
• 期：4
• 页码：493-509
• 全文大小：481 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：None Assigned
• 出版者：Birkh盲user Basel
• ISSN：1662-999X
• 卷排序：7

文摘

In the present paper, we extend in the first place the main results of Tretter in (Spectral theory block operator matrices and applications. Imperial College Press, London, 2008) to linear relations. Moreover, we define a matrix linear relation. We denote by \(\mathcal {L}\) the block matrix linear relation, acting on the Banach space \(X\oplus Y\), of the form $$\begin{aligned} \mathcal {L}= \left( \begin{array}{ll} A &{} B\\ C &{} D \\ \end{array} \right) , \end{aligned}$$where A, B, C and D are four closable linear relations with dense domains. For diagonally dominant and off-diagonally dominant of block matrix linear relation \(\mathcal {L}\), we give a necessary and sufficient condition for \(\mathcal {L}\) to become closed and closable. In the second place, we study the stability of the essential spectrum of this block linear relation.KeywordsMatrix of linear relationsDiagonally dominant operatorsMathematics Subject Classification47A06References1.Ammar, A., Diagana, T., Jeribi, A.: Perturbations of Fredholm linear relations in Banach spaces with application to \(3 \times 3\)-block matrices of linear relations. Arab. J. Math. Sci. 22(1), 59–76 (2015)MathSciNetCrossRefMATHGoogle Scholar2.Aymen, A., Mohammed Zerai, D., Jeribi, A.: Some properties of upper triangular \(3\times 3\)-block matrices of linear relations. Boll. Unione Mat. Ital. 8(3), 189–204 (2015)3.Agarwal, R.P., Meehan, R.P., O’Regan, M.: Fixed Points Theory and Applications. Cambridge University Press, Cambridge (2001)CrossRefMATHGoogle Scholar4.Álvarez, T., Ammar, A., Jeribi, A.: A characterization of some subsets of \(\cal {S}\)-essential spectra of a multivalued linear operator. Colloq. Math. 135(2), 171–186 (2014)5.Álvarez, T., Ammar, A., Jeribi, A.: On the essential spectra of some matrix of linear relations. Math. Methods Appl. Sci. 37, 620–644 (2014)MathSciNetCrossRefMATHGoogle Scholar6.Álvarez, T., Cross, R.W., Wilcox, D.: Quantities related to upper and lower semi-Fredholm type relations. Bull. Aust. Math. Soc. 66, 275–289 (2002)MathSciNetCrossRefMATHGoogle Scholar7.Álvarez, T., Cross, R.W., Wilcox, D.: Multivalued Fredholm type operators with abstract generalised inverses. J. Math. Anal. Appl. 261, 403–417 (2001)8.Coddington, E.A.: Multivalued Operators and Boundary Valued Problems . Lecture Notes in Math, vol. 183. Springer-Verlag, Berlin (1971)9.Cross, R.W.: Multivalued Linear Operators. Marcel Dekker Inc., New York (1998)10.Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, Oxford Science Publications (1987)11.Elleuch, S., Mnif, M.: Essential approximate point spectra for upper triangular matrix of linear relations. Acta Math. 33(4) 1187–1201 (2013)12.Favini, A., Yagi, A.: Multivalued linear operators and degenerate evolution equations. Ann. Mat. Pura Appl. 163, 353–384 (1993)MathSciNetCrossRefMATHGoogle Scholar13.Gorniewicz, L.: Topological Fixed Point theory and Multivalued Mappings. Kluwer, New York (1999)14.Gromov, M.: Partial Differential Relations. Springer-Verlag, Berlin (1986)CrossRefMATHGoogle Scholar15.Jeribi, A.: Spectral Theory and Applications of Linear Operators and Block Operator Matrices. Springer-Verlag, New York (2015)CrossRefMATHGoogle Scholar16.Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Springer, Berlin (1995) (Reprint of the edition (1980))17.Muresan, M.: On a boundary value problem for quasi-linear differential inclusions of evolution. Collect. Math. 45, 165–175 (1994)MathSciNetMATHGoogle Scholar18.Roman-Flores, H., Flores- Franulic, A., Rojas-Medar, M.A., Bassanezi, R.C.: Stability of the fixed points set of fuzzy contractions. Appl. Math. Lett. 11, 33–37 (1988)19.Tretter, C.: Spectral Theory Block Operator Matrices and Applications. Imperial College Press, London (2008)20.Von Neumann, J.: Functional Operatos II, The Geometry of orthogonal spaces. Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (1950)21.Wilcox, D.: Multivalued Semi-Fradholm Operators in Normed Linear Spaces A thesis submitted in fulfilment of the requirements for the degree oh PhD in Mathematics. Departement of Mathematics and Applied Mathematics, University of Cape Town, December (2012)Copyright information© Springer International Publishing 2016Authors and AffiliationsAymen Ammar1Slim Fakhfakh1Aref Jeribi1Email author1.Département de MathématiquesUniversité de Sfax, Faculté des Sciences de SfaxSfaxTunisia About this article CrossMark Print ISSN 1662-9981 Online ISSN 1662-999X Publisher Name Springer International Publishing About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips