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Abstract

The theory of measures of noncompactness and measures of weak noncompactness has many applications in topology, functional analysis, and operator theory. In this chapter, we consider one axiomatic approach to this notion which includes the most important classical definitions. We give some results concerning a certain class of semi-Fredholm and Fredholm operators via the concept of measures of noncompactness and measures of weak noncompactness.

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Bibliography

  1. B. Abdelmoumen, Stabilité des spectres essentiels et applications à des modéles cinétiques, Thesis, University of Sfax, 2010

    Google Scholar 

  2. B. Abdelmoumen, H. Baklouti, Fredholm perturbations and seminorms related to upper semi-Fredholm perturbations. Filomat 27(6), 1147–1155 (2013)

    MathSciNet  Google Scholar 

  3. B. Abdelmoumen, O. Jedidi, A. Jeribi, Time asymptotic description of an abstract Cauchy problem’s solution and application to transport equation. Appl. Math. 59(1), 53–67 (2014)

    MathSciNet  MATH  Google Scholar 

  4. B. Abdelmoumen, A. Jeribi, M. Mnif, Time asymptotic description of the solution to an abstract Cauchy problem and application to transport equation. Math. Z. 268(3–4), 837–869 (2011)

    MathSciNet  MATH  Google Scholar 

  5. B. Abdelmoumen, A. Jeribi, M. Mnif, Invariance of the Schechter essential spectrum under polynomially compact operators perturabation. Extracta Math. 26(1), 61–73 (2011)

    MathSciNet  MATH  Google Scholar 

  6. B. Abdelmoumen, A. Jeribi, M. Mnif, On graph measures in Banch spaces and description of essential spectra of multidimensional transport equation. Acta Math. Sci. Ser. B Engl. Ed. 32(5), 2050–2064 (2012)

    MathSciNet  MATH  Google Scholar 

  7. B. Abdelmoumen, A. Jeribi, M. Mnif, Measure of weak noncompactness, some new properties in Fredholm theory, characterization of the Schechter essential spectrum and application to transport operators. Ricerche Mat. 61, 321–340 (2012)

    MathSciNet  Google Scholar 

  8. B. Abdelmoumen, A. Dehici, A. Jeribi, M. Mnif, Some new properties of Fredholm theory, essential spectra and application to transport theory. J. Inequal. Appl. 2008, 1–14 (2008)

    MathSciNet  Google Scholar 

  9. F. Abdmouleh, Fredholm operators, essential spectra of sum of two bounded linear operators and applications to a transport operators, Thesis, University of Sfax, 2009

    Google Scholar 

  10. F. Abdmouleh, A. Ammar and A. Jeribi, A Characterization of the pseudo-Browder essential spectra of linear operators and application to a transport equation, J. Comp. Theo. Tran., DOI: 10.1080/23324309.2015.1033222 (2015)

  11. F. Abdmouleh, A. Jeribi, Symmetric family of Fredholm operators of indices zero, stability of essential spectra and application to transport operators. J. Math. Anal. Appl. 364, 414–423 (2010)

    MathSciNet  MATH  Google Scholar 

  12. F. Abdmouleh, A. Jeribi, Gustafson, Weidman, Kato, Wolf, Schechter, Browder, Rakoc̆ević and Schmoeger essential spectra of the sum of two bounded operators and application to a transport operator. Math. Nachr. 284(2–3), 166–176 (2011)

    MathSciNet  MATH  Google Scholar 

  13. F. Abdmouleh, A. Ammar, A. Jeribi, Stability of the S-essential spectra on a Banach space. Math. Slovaca 63(2), 299–320 (2013)

    MathSciNet  MATH  Google Scholar 

  14. F. Abdmouleh, S. Charfi, A. Jeribi, On a characterization of the essential spectra of the sum and the product of two operators. J. Math. Anal. Appl. 386(1), 83–90 (2012)

    MathSciNet  MATH  Google Scholar 

  15. S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math. 15, 119–147 (1962)

    MathSciNet  MATH  Google Scholar 

  16. M.S. Agranovich, M.I. Vishik, Elliptic problems with parameter and parabolic problems of a general form. Uspekhi Matem. Nauk 19, 53–161 (1964) (Russian)

    MATH  Google Scholar 

  17. P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers (Kluwer Academic, Dordrecht, 2004)

    MATH  Google Scholar 

  18. R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina, B.N. Sadovskii, Measures of Noncompactness and Condensing Operators (Birkhäuser, Basel, 1992)

    MATH  Google Scholar 

  19. C.D. Aliprantis, O. Burkinshaw, Positive Operators (Academic Press, Orlando, 1985)

    MATH  Google Scholar 

  20. A. Ammar, S-essential spectra, the Weyl pseudospectra of linear operators, perturbation theory of semi regular and essentially semi regular operators, Thesis, University of Sfax, 2013

    Google Scholar 

  21. A. Ammar, A. Jeribi, A characterization of the essential pseudo-spectra on a Banach space. Arab. J. Math. 2(2), 139–145 (2013)

    MathSciNet  MATH  Google Scholar 

  22. A. Ammar, A. Jeribi, A characterization of the essential pseudo-spectra and application to a transport equation. Extracta Math. 28(1), 95–112 (2013)

    MathSciNet  MATH  Google Scholar 

  23. A. Ammar, A. Jeribi, Measures of noncompactness and essential pseudo-spectra on Banach Space. Math. Methods Appl. Sci. 37(3), 447–452 (2014)

    MathSciNet  MATH  Google Scholar 

  24. A. Ammar, B. Boukattaya, A. Jeribi, Stability of the S-left and S-right essential spectra of a linear operator. Acta Math. Sci. 34B(5), 1–13 (2014)

    Google Scholar 

  25. A. Ammar, A. Jeribi, N. Moalla, On a characterization of the essential spectra of a 3 × 3 operator matrix and application to three-group transport operators. Ann. Funct. Anal. 4(2), 153–170 (2013) (electronic only)

    Google Scholar 

  26. F. Andreu, J. Martinez, J.M. Mazon, A spectral mapping theorem for perturb ed strongly continuous semigroup. Math. Ann. 291, 453–462 (1991)

    MathSciNet  MATH  Google Scholar 

  27. N. Angelescu, N. Marinescu, V. Protopopescu, Linear monoenergetic transport with reflecting boundary conditions. Rev. Roum. Phys. 19, 17–26 (1974)

    Google Scholar 

  28. N. Angelescu, N. Marinescu, V. Protopopcu, Neutron transport with periodic boundary conditions. lkansp. Theor. Stat. Phys. 5, 115–125 (1976)

    Google Scholar 

  29. C. Anné, N. Torki-Hamza, The Gauß-Bonnet operator of infinite graph, preprint, arXiv: 1301.0739 (2013)

    Google Scholar 

  30. P.M. Anselone, Collectively Compact Operator Approximation Theory (Prentice-Hall, Englewood Cliffs, 1971)

    MATH  Google Scholar 

  31. C. Apostol, The reduced minimum modulus. Mich. Math. J. 32, 279–294 (1985)

    MathSciNet  MATH  Google Scholar 

  32. K. Appel, W. Haken, Every planar map is four, clorable, Part I. Discharging. lllinois J. Math. 21, 429–490 (1977)

    MathSciNet  MATH  Google Scholar 

  33. K. Appel, W. Haken, Every planar map is four clorable. Part II. Reducibility. lllinois J. Math. 21, 491–567 (1977)

    MathSciNet  MATH  Google Scholar 

  34. W. Arendt, Resolvent positive operators. Proc. Lond. Math. Soc. 54, 321–349 (1987)

    MathSciNet  MATH  Google Scholar 

  35. W. Arend, R. Nagel (ed.), One-Parameter Semigroups of Positive Operators. Lecture Notes in Mathematics, vol. 1184 (Springer, Heidelberg, 1986)

    Google Scholar 

  36. Z. Artstein, Continuous dependence of solutions of operator equations, I. Trans. Am. Math. Soc. 231(1), 143–166 (1977)

    MathSciNet  MATH  Google Scholar 

  37. K. Astala, On measure of noncompactness and ideal variations in Banach spaces. Ann. Acad. Sci. Fenn. Ser. A. I. Math. Diss. 29, (1980)

    Google Scholar 

  38. K. Astala, H.-O. Tylli, Seminorms related to weak compactness and to Tauberian operators. Math. Proc. Camb. Philos. Soc. 107, 367–375 (1990). Printed in Great Britain

    Google Scholar 

  39. F.V. Atkinson, The normal solubility of linear equations in normed spaces. Math. Sb. (N.S) 28(70), 3–14 (1951) (Russian)

    Google Scholar 

  40. F.V. Atkinson, H. Langer, R. Mennicken, A.A. Shkalikov, The essential spectrum of some matrix operators. Math. Nachr. 167, 5–20 (1994)

    MathSciNet  MATH  Google Scholar 

  41. B. Aupetit, A Primer on Spectral Theory (Springer, New York, 1991)

    MATH  Google Scholar 

  42. J.M. Ayerbe Toledano, T. Dominguez Benavides, G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory (Birkhäuser, Basel, 1997)

    MATH  Google Scholar 

  43. G. Ball, Diffusion approximation of the radiative transfer equations in a chanel. Trans. Theor. Stat. Phys. 30(2 & 3), 269–293 (2001)

    Google Scholar 

  44. H. Baloudi, S. Golénia, A. Jeribi, The adjacency matrix and the discrete Laplacian acting on forms, (preprint) (2015)

    Google Scholar 

  45. H. Baloudi, A. Jeribi, Left-Right Fredholm and Weyl spectra of the sum of two bounded operators and application. Mediterr. J. Math. 11, 939–953 (2014)

    MathSciNet  MATH  Google Scholar 

  46. J. Banaś, Applications of measure of weak noncompactness and some classes of operators in the theory of functional equations in the Lebesgue space, in Proceedings of the second World Congress of Nonlinear Analysis, Part 6, Athen, 1966. Nonlinear Anal. 30, 3283–3293 (1997)

    MathSciNet  MATH  Google Scholar 

  47. J. Banaś, K. Geobel, Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60 (Marcel Dekker, New York, 1980), pp. 259–262.

    Google Scholar 

  48. J. Banaś, A. Martinón, On measures of weak noncompactness in Banach sequence spaces. Portugal. Math. 52, 131–138 (1995)

    MathSciNet  MATH  Google Scholar 

  49. J. Banaś, J. Rivero, On measures of weak noncompactness. Ann. Mat. Pura Appl. 151, 213–262 (1988)

    MathSciNet  MATH  Google Scholar 

  50. A. Bátkai, P. Binding, A. Dijksma, R. Hryniv, H. Langer, Spectral problems for operator matrices. Math. Nachr. 278, 1408–1429 (2005)

    MathSciNet  MATH  Google Scholar 

  51. R. Beals, V. Protopopescu, Abstract time dependent transport equations. J. Math. Anal. Appl. 121, 370–405 (1987)

    MathSciNet  MATH  Google Scholar 

  52. L.W. Beineke, Derived graphs and digraphs, in Beitrage zur Graphentheorie, ed. by H. Sachs, H. Voss, H. Walther (Tenbner, Leipzig, 1968), pp. 17–33

    Google Scholar 

  53. A. Belleni-Morante, Neutron transport in a nonuniform slab with generalized boundary conditions. J. Math. Phys. 11, 1553–1558 (1970)

    MathSciNet  MATH  Google Scholar 

  54. M. Belzad, A criterion for the planarity of a total garaph. Proc. Camb. Philos. Soc. 63, 679–681(1967)

    Google Scholar 

  55. N. Ben Ali, Base de Riesz de vecteurs propres d’une famille d’opérateurs, spectres essentiels d’un opérateur matriciel et applications, Thesis, University of Sfax, 2011

    Google Scholar 

  56. N. Ben Ali, A. Jeribi, N. Moalla, Essential spectra of some matrix operators. Math. Nachr. 283(9), 1245–1256 (2010)

    MathSciNet  MATH  Google Scholar 

  57. A. Ben Amar, Spectral and fixed point theories and applications to problems arising in kinetic theory of gas and in growing cell populations, Thesis, University of Sfax, 2007

    Google Scholar 

  58. A. Ben Amar, A. Jeribi, M. Mnif, Some applications of the regularity and irreducibility on transport theory. Acta Appl. Math. 110, 431–448 (2010)

    MathSciNet  MATH  Google Scholar 

  59. A. Ben Amar, A. Jeribi, B. Krichen, Essential spectra of a 3x3 operator matrix and application to three-group transport equation. Integr. Equ. Oper. Theory 68, 1–21 (2010)

    MathSciNet  MATH  Google Scholar 

  60. A. Ben Amar, A. Jeribi, M. Mnif, Some results on Fredholm and semi-Fredholm operators. Arab. J. Math. 3(3), 313–323 (2014)

    MathSciNet  Google Scholar 

  61. M. Benharrat, A. Ammar, A. Jeribi, B. Messirdi, On the Kato, semi-regular and essentially semi-regular spectra. Funct. Anal. Approx. Comput. 6(2), 9–22 (2014)

    MathSciNet  Google Scholar 

  62. M. Berkani, A. Ouahab, Opérateur essentiellement régulier dans les espaces de Banach. Rend. Circ. Math. Palermo Serie II 46, 131–160 (1997)

    MathSciNet  MATH  Google Scholar 

  63. N. Biggs, E. Lioyd, R. Wilson, Graph Theory (Oxford University Press, Oxford, 1986), pp. 1736–1936

    MATH  Google Scholar 

  64. P. Binding, R. Hryniv, Relative boundedness and relative compactness for linear operators in Banach spaces. Proc. Am. Math. Soc. 128, 2287–2290 (2000)

    MathSciNet  MATH  Google Scholar 

  65. G. Borgioli, S. Totaro, On the spectrum of the transport operator with mixed type boundary conditions, in Atti Congruso, Aimeta, vol. 1 (1986), pp. 393–398

    Google Scholar 

  66. F.E. Browder, On the spectral theory of elliptic differential operators, I. Math. Ann. 142, 22–130 (1961)

    MathSciNet  MATH  Google Scholar 

  67. S.R. Caradus, Operators of Riesz type. Pac. J. Math. 18, 61–71 (1966)

    MathSciNet  MATH  Google Scholar 

  68. S.R. Caradus, W.E. Plaffenberger, B. Yood, Calking Algebras and Algebras of Operators on Banach Spaces. Lecture Notes, vol. 9 (Marcel Dekker, New York, 1974)

    Google Scholar 

  69. R. Carlson, Adjoint and self-adjoint differential operators on graphs. J. Differ. Equ. 6, 1–10 (1998)

    Google Scholar 

  70. A.L. Cauchy, Recherche sur les polyodres premier mémoire. Journal de l’école Polytechnique 9, 66–86 (1813)

    Google Scholar 

  71. A. Cayley, On the theory of the analytical forms called trees. Philos. Mag. 13, 172–176 (1857)

    Google Scholar 

  72. A. Cayley, Ueber die Analytischen Figuren, welche in der Mathematic Baume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Berichteder deutshen chemischen Gesellsoft 8(2), 1056–1059 (1875)

    MathSciNet  Google Scholar 

  73. W. Chaker, A. Jeribi, B. Krichen, Demicompact linear operators, essential spectrum and some perturbation results. Math. Nachr. 1–11 (2015). doi:10.1002/mana.201200007

  74. S. Charfi, Spectral properties of operator matrices, perturbed linear operators, systems of evolution equations and applications, Thesis, University of Sfax, 2010

    Google Scholar 

  75. S. Charfi, On the time asymptotic behavior of a transport operator with diffuse reflection boundary condition. Transp. Theory Stat. Phys. 41(7), 529–551 (2012)

    MathSciNet  MATH  Google Scholar 

  76. S. Charfi, A. Jeribi, On a characterization of the essential spectra of some matrix operators and applications to two-group transport operators. Math. Z. 262(4), 775–794 (2009)

    MathSciNet  MATH  Google Scholar 

  77. S. Charfi, A. Jeribi, N. Moalla, Time asymptotic behavior of the solution of an abstract Cauchy problem given by a one-velocity transport operator with Maxwell boundary condition. Collect. Math. 64, 97–109 (2013)

    MathSciNet  MATH  Google Scholar 

  78. S. Charfi, A. Jeribi, R. Moalla, Essential spectra of operator matrices and applications. Methods Appl. Sci. 37(4), 597–608 (2014)

    MathSciNet  MATH  Google Scholar 

  79. S. Charfi, A. Jeribi, I. Walha, Essential spectra, matrix operator and applications. Acta Appl. Math. 111(3), 319–337 (2010)

    MathSciNet  MATH  Google Scholar 

  80. F.R.K. Chung, Spectral Graph Theory. CBMS Regional Conferance Series in Mathematics, vol. 92 (American Mathematical Society, Providence, 1997), xi, 207 pp.

    Google Scholar 

  81. Ph. Clement, One-Parameter Semigroups (North-Holland, Amsterdam, 1987)

    MATH  Google Scholar 

  82. Y. Colin de Verdiére, Spectres de graphes, in Cours Spécialisés, vol. 4 (Société Mathématique de France, Paris, 1998)

    Google Scholar 

  83. A. Corciovei, V. Protopopescu, On the spectrum of the linear transport oper ator with diffuse reflections. Rev. Roum. Phys. 21, 713–719 (1976)

    MathSciNet  Google Scholar 

  84. J.R. Cuthbert, On semigroups such that U(t) − I is compact for some t > 0. Z. Wahrschein- lichkeitstheorie und Verw. Gebiete 18, 9–16 (1971)

    MathSciNet  Google Scholar 

  85. D. Cvetković, On gaps between bounded operators. Publ. Inst. Math. 72(86), 49–54 (2002)

    Google Scholar 

  86. M. Damak, A. Jeribi, On the essential spectra of some matrix operators and applications. Electron. J. Differ. Equ. 11, 1–16 (2007)

    MathSciNet  Google Scholar 

  87. J. Danes, On the Istratescu measure of noncompactness. Bull. Math. Soc. R. S. Roum. 16(64), 403–406 (1972)

    MathSciNet  Google Scholar 

  88. E.B. Davies, Spectral Theory and Differential Operators (Cambridge University Press, Cambridge, 1996)

    Google Scholar 

  89. R. Dautray, J.L. Lions, Analyse Mathématique et Calcul Numérique, vol. 9 (Masson, Paris, 1988)

    Google Scholar 

  90. F.S. De Blasi, On a property of the unit sphere in a Banach spaces. Bull. Math. Soc. Sci. Math. R. S. Roum. 21(69), 259–262 (1977)

    Google Scholar 

  91. S. Degong, Some notes on the spectral properties of C 0-semigroups generated by linear transport operators. Trans. Theor. Stat. Phys. 26(1–2), 233–242 (1997)

    MATH  Google Scholar 

  92. A. Dehici, K. Latrach, A. Jeribi, On a transport operator arising in growing cell populations. Spectral analysis. Adv. Math. Res. 1, 159–185 (2002) (Nova Sci. Publ., Hauppauge)

    Google Scholar 

  93. A. Dehici, A. Jeribi, K. Latrach, Spectral analysis of a transport operator arising in growing cell populations. Acta Appl. Math. 92(1), 37–62 (2006)

    MathSciNet  MATH  Google Scholar 

  94. J. Diestel, Geometry of Banach Spaces-Selected Topics. Lecture Notes in Mathematics, vol. 485 (Springer, New York, 1975)

    Google Scholar 

  95. J. Diestel, A survey of results related to Dunford-Pettis property, in Cont. Math.2, Amer. Math. Soc. of Conf. on Integration, Topology and Geometry in Linear Spaces (1980), pp. 15–60

    Google Scholar 

  96. P. Dodds, D.H. Fremlin, Compact operators in Banach lattices. Isr. J. Math. 34, 287–320 (1979)

    MathSciNet  MATH  Google Scholar 

  97. T. Dominguez Benavides, Some properties of the set and ball measures of noncompactness and applications. J. Lond. Math. Soc. 34(2), 120–128 (1986)

    MathSciNet  MATH  Google Scholar 

  98. R. Drnov\(\check{s}\) ek, Bounds for the spectral radius of positive operators. Comment. Math. Univ. Carol. 41(3), 459–467 (2000)

    Google Scholar 

  99. J.J. Duderstart, W.R. Martin, Transport Theory (Willey, New York, 1979)

    Google Scholar 

  100. N. Dunford, B.J. Pettis, Linear operations on summable functions. Trans. Am. Math. Soc. 47, 323–392 (1940)

    MathSciNet  Google Scholar 

  101. N. Dunford, J.T. Schwartz, Linear Operators, Part I. General Theory (Interscience, New York, 1958)

    Google Scholar 

  102. D.E. Edmum, W.D. Evans, Spectral Theory and Differential Operators (Oxford Science Publications, Oxford, 1987)

    Google Scholar 

  103. Y. Eidelman, V. Milman, A. Tsolomitis, Functional Analysis, Graduate. Studies in Mathematics, vol. 66 (American Mathematical Society, Providence, 2004) (An introduction)

    Google Scholar 

  104. G. Emmanuele, Measure of weak non compactness and fixed point theorems. Bull. Math. Soc. Sci. Math. R.S. Roum. 25, 353–358 (1981)

    Google Scholar 

  105. K.J. Engel, Positivity and stability for one-sided coupled operator matrices. Positivity 1, 103–124 (1997)

    MathSciNet  MATH  Google Scholar 

  106. K.J. Engel, R. Nagel, One-Parameters Semigroup for Linear Evolutions Equations. Graduate text in Mathematics (Springer, New York, 2000)

    Google Scholar 

  107. I.D. Evzerov, Domains of fractional powers of ordinary differential operators in L p -spaces. Math. Zametki (Engl. Transl. in Math. Notes) 21(4), 509–518 (1977)

    Google Scholar 

  108. I.D. Evzerov, P.E. Sobolevskii, Fractional powers of ordinary differential operators. Differencial’nye Uravnenija 9, 228–240 (1973)

    MathSciNet  Google Scholar 

  109. P. Exner, J. Keating, P. Kuchment, T. Sunada, A. Teplyaev, Analysis on Graphs and Its Applications (American Mathematical Society, Providence, 2008)

    MATH  Google Scholar 

  110. M. Faierman, R. Mennicken, M. Möller, The essential spectrum of a system of singular ordinary differential operators of mixed order. Part I: The general problem and an almost regular case. Math. Nachr. 208, 101–115 (1999)

    MATH  Google Scholar 

  111. F. Fakhfakh, M. Mnif, Perturbation of semi-Browder operators and stability of Browder’s essential defect and approximate point spectrum. J. Math. Anal. Appl. 347(1), 235–242 (2008)

    MathSciNet  MATH  Google Scholar 

  112. J.M.G. Fell, R.S. Doran, Representations of *-Algebras Locally compact Groups, and Banach *-Algebraic Bundles, vol. 1. Basic Representation Theory of Groups and Algebras. Pure Appl. Math., vol. 125 (Academic Press, Boston 1988)

    Google Scholar 

  113. I. Fredholm, Sur une classe d’équations fonctionelles. Acta Math. 27, 365–390 (1903)

    MathSciNet  MATH  Google Scholar 

  114. M. Garden, Fractal Music, Hypercads, and More Mathematical Recreations from Scientific American (W. H. Freeman and Company, San Francisco, 1992), p. 203

    Google Scholar 

  115. V. Georgescu, S.Golénia, Compact perturbations and stability of the essential spectrum of singular differential operators. J. Oper. Theory 59, 115–155 (2008)

    MATH  Google Scholar 

  116. I. Ghoberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, vol. 1 (Birkhäuser, Basel, 1990)

    Google Scholar 

  117. F. Gilfeather, The structure and asymptotic behavior of polynomially compact operators. Proc. Am. Math. Soc. 25, 127–134 (1970)

    MathSciNet  MATH  Google Scholar 

  118. S.K. Godunov, V.S. Ryabenki, Theory of Difference Schemes, an Introduction. Translated by E. Godfredsen (North-Holland, Amsterdam; Interscience Publishers, New York; Wiley, New York, 1964)

    Google Scholar 

  119. I.C. Gohberg, On linear equations in Hilbert space. Dokl. Akad. Nauk SSSR (N.S.) 76, 9–12 (1951) (Russian)

    Google Scholar 

  120. I.C. Gohberg, On linear equations in normed spaces. Dokl. Akad. Nauk SSSR (N.S.) 76, 477–480 (1951) (Russian)

    Google Scholar 

  121. I.C. Gohberg, On linear operators depending analytically on a parameter. Dokl. Akad. Nauk SSSR (N.S.) 78, 629–632 (1951) (Russian)

    Google Scholar 

  122. I.C. Gohberg, On the index of an unbounded operator. Mat. Sb. (N.S) 33(75), 193–198 (1951) (Russian)

    Google Scholar 

  123. I.C. Gohberg, G. Krein, Fundamental theorems on deficiency numbers, root numbers and indices of linear operators. Am. Math. Soc. Transl. Ser. 2 13, 185–264 (1960)

    MathSciNet  Google Scholar 

  124. I.C. Gohberg, A.S. Markus, I.A. Feldman, Normally solvable operators and ideals associated with them. Am. Math. Soc. Transl. Ser. 2, 61, 63–84 (1967)

    Google Scholar 

  125. I.C. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators Vol. I. Operator Theory: Advances and Applications, vol. 49 (Birkhauser, Basel, 1990)

    Google Scholar 

  126. S. Goldberg, Unbounded Linear Operators (McGraw-Hill, New York, 1966)

    MATH  Google Scholar 

  127. S. Goldberg, Perturbations of semi-Fredholm operators by operators converging to zero compactly. Proc. Am. Math. Soc. 45(1), 93–98 (1974)

    MATH  Google Scholar 

  128. M.A. Goldman, S.N. Krackovskii, Behaviour of the space of zero elements with finite-dimensional salient on the Riesz kernel under perturbations of the operator. Dokl. Akad. Nauk SSSR 221, 532–534 (1975); English transl., Soviet Math. Dokl. 16, 370–373 (1975)

    Google Scholar 

  129. S. Golénia, Unboundedness of adjacency matrices of locally finite graphs. Lett. Math. Phys. 93, 127–140 (2010)

    MathSciNet  MATH  Google Scholar 

  130. S. Golénia, Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians. J. Funct. Anal. 266, 2662–2688 (2014)

    MathSciNet  MATH  Google Scholar 

  131. S. Golénia, C. Schumacher, The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs. J. Math. Phys. 52, 063512, 17 pp. (2011)

    Google Scholar 

  132. M. Gonzàlez, A. Martinon, On the generalized Sadovskii functor. Rev. Acad. Canaria Cien. 1, 109–117 (1990)

    MATH  Google Scholar 

  133. M. Gonzàlez, E. Saksman, H.-O. Tylli, Representing non-weakly compact operators. Stud. Math. 113, 265–282 (1995)

    MATH  Google Scholar 

  134. W.T. Gowers, A solution to the Schroeder-Bernstein problem for Banach spaces. Bull. Lond. Math. Soc. 28, 297–304 (1996)

    MathSciNet  MATH  Google Scholar 

  135. W.T. Gowers, B. Maurey, The unconditional basic sequence problem. J. Am. Math. Soc. 6, 851–874 (1993)

    MathSciNet  MATH  Google Scholar 

  136. S. Grabiner, Ascent, descent, and compact perturbations. Proc. Am. Math. Soc. 71, 79–80 (1978)

    MathSciNet  MATH  Google Scholar 

  137. B. Gramsch, D. Lay, Spectral mapping theorems for essential spectra. Math. Ann. 192, 17–32 (1971)

    MathSciNet  MATH  Google Scholar 

  138. W. Greenberg, C. Van der Mee, V. Protopopescu, Boundary Value Problems in Abstract Kinetic Theory (Birkhäuser, Basel, 1987)

    MATH  Google Scholar 

  139. G. Greiner, Spectral properties and asymptotic behavior of the linear transport equation. Math. Z. 185, 167–177 (1984)

    MathSciNet  MATH  Google Scholar 

  140. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24 (Pitman, Boston, 1985)

    Google Scholar 

  141. J.J. Grobler, A note on the theorems of Jentzsch-Perron and Frobenius. Indagationes Math. 49, 381–391 (1987)

    MathSciNet  MATH  Google Scholar 

  142. J.J. Grobler, Spectral theory on Banach lattice, in Operator Theory in Function Spaces and Banach Lattice. Oper. Theory Adv. Appl., vol. 75 (Birkhäuser, Basel, 1995), pp. 133–172

    Google Scholar 

  143. A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type C(K). Can. J. Math. 5, 129–173 (1953)

    MathSciNet  MATH  Google Scholar 

  144. K. Gustafson, J. Weidmann, On the essential spectrum. J. Math. Anal. Appl. 25, 121–127 (1969)

    MathSciNet  MATH  Google Scholar 

  145. P.R. Halmos, V.S. Sunder, Bounded integral operator on L 2 spaces, in Ergebnisses der Mathematic und ihrer Grenzgebiete. Results in Mathematics and Related Areas, vol. 96 (Springer, Berlin, 1978)

    Google Scholar 

  146. Y.M. Han, S.H. Lee, W.Y. Lee, On the structure of polynomially compact opertors. Math. Z. 232, 257–263 (1999)

    MathSciNet  MATH  Google Scholar 

  147. V. Hardt, Uber ein im eigenwertparameter rationales randeigenwertproblem bei differentialgleichungssystemen zweiter ordnung, Dissertation, Regensburg, 1992

    Google Scholar 

  148. A. Harrabi, Pseudospectrum of a sequence of bounded operators. RAIRO Modél. Math. Anal. Numér. 32(6), 671–680 (1998)

    MathSciNet  MATH  Google Scholar 

  149. H. Heesch, Untersuchumgen zum weirfarbenproblem Mannhein. Bibliographiscles Institut (1969)

    Google Scholar 

  150. H.J.A.M. Heijmans, Structured populations, linear semigroups and positivity. Math. Z. 191, 599–617 (1986)

    MathSciNet  MATH  Google Scholar 

  151. H. Henriquez, Cosine operator families such that C(t) − I is compact for all t > 0. Indian J. Pure Appl. Math. 16, 143–152 (1985)

    MathSciNet  MATH  Google Scholar 

  152. E. Hille, R.S. Phillips, Functional Analysis and Semigroups, vol. 31 (American Mathematical Society Colloquium Publications, Rhode Island, 1957)

    MATH  Google Scholar 

  153. P.D. Hislop, I.M. Segal, Introduction to STheory with Applications to Schrodinger Operators (Springer, New York, 1966)

    Google Scholar 

  154. S. Huillier, Mémoire sur la polyédrométrie. Annales de Mathématiques 3, 169–189 (1861)

    Google Scholar 

  155. V.I. Istratescu, Some remarks on a class of semigroups of operators. Z. Wahrscheinlichkeit- stheorie und Verw. Gebiete 26, 241–243 (1973)

    MathSciNet  MATH  Google Scholar 

  156. V.I. Istrateescu, Introduction to Linear Operator Theory (Mareel Dekker, New York, 1981)

    Google Scholar 

  157. O. Jedidi, Spectral theory of C 0-semigroups and stability of some essential spectra of linear relations on Banach spaces, Thesis, University of Sfax, 2013

    Google Scholar 

  158. A. Jeribi, Quelques remarques sur les opérateurs de Fredholm et application à l’équation de transport. C. R. Acad. Sci. Paris Sér. I 325, 43–48 (1997)

    MathSciNet  MATH  Google Scholar 

  159. A. Jeribi, Quelques remarques sur le spectre de Weyl et applications. C. R. Acad. Sci. Paris Sér. I 327, 485–490 (1998)

    MathSciNet  MATH  Google Scholar 

  160. A. Jeribi, Développement de certaines propriétés fines de la théorie spectrale et applications à des modèles monocinétiques et à des modèles de Reggeons, Thesis of Mathematics, University of Corsica, Frensh, 16 Janvier 1998

    Google Scholar 

  161. A. Jeribi, Une nouvelle caractérisation du spectre essentiel et application. C. R. Acad. Sci. Paris Sér. I 331, 525–530 (2000)

    MathSciNet  MATH  Google Scholar 

  162. A. Jeribi, A characterization of the essential spectrum and applications. Boll. dell. Unio. Mate. Ital. 8 B-5, 805–825 (2002)

    Google Scholar 

  163. A. Jeribi, A characterization of the Schechter essential spectrum on Banach spaces and applications. J. Math. Anal. Appl. 271, 343–358 (2002)

    MathSciNet  MATH  Google Scholar 

  164. A. Jeribi, Some remarks on the Schechter essential spectrum and applications to transport equations. J. Math. Anal. Appl. 275, 222–237 (2002)

    MathSciNet  MATH  Google Scholar 

  165. A. Jeribi, On the Schechter essential spectrum on Banach spaces and applications. Ser. Math. Inf. 17, 35–55 (2002)

    MathSciNet  MATH  Google Scholar 

  166. A. Jeribi, Time asymptotic behavior for unbounded linear operator arising in growing cell populations. Nonlinear Anal. Real World Appl. 4, 667–688 (2003)

    MathSciNet  MATH  Google Scholar 

  167. A. Jeribi, Fredholm operators and essential spectra. Arch. Inequal. Appl. 2(2–3), 123–140 (2004)

    MathSciNet  MATH  Google Scholar 

  168. A. Jeribi, K. Latrach, Quelques remarques sur le spectre essentiel et application à l’équation de transport. C. R. Acad. Sci. Paris Sér. I 323, 469–474 (1996)

    MathSciNet  MATH  Google Scholar 

  169. A. Jeribi, K. Latrach, H. Megdiche, Time asymptotic behavior of the solution to a Cauchy problem governed by a transport operator. J. Integral Equ. Appl. 17(2), 121–139 (2005)

    MathSciNet  MATH  Google Scholar 

  170. A. Jeribi, M. Mnif, Fredholm operators, essential spectra and application to transport equation. Acta Appl. Math. 89, 155–176 (2005)

    MathSciNet  MATH  Google Scholar 

  171. A. Jeribi, N. Moalla, Fredholm operators and Riesz theory for polynomially compact operators. Acta Appl. Math. 90(3), 227–245 (2006)

    MathSciNet  MATH  Google Scholar 

  172. A. Jeribi, N. Moalla, A characterization of some subsets of Schechter’s essential spectrum and application to singular transport equation. J. Math. Anal. Appl. 358, 434–444 (2009)

    MathSciNet  MATH  Google Scholar 

  173. A. Jeribi, I. Walha, Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectra of some matrix operator and application to two-group transport equation. Math. Nachr. 284(1), 67–86 (2011)

    MathSciNet  MATH  Google Scholar 

  174. A. Jeribi, H. Megdiche, N. Moalla, On a transport operator arising in growing cell populations II. Cauchy problem. Math. Methods Appl. Sci. 28, 127–145 (2005)

    MathSciNet  MATH  Google Scholar 

  175. A. Jeribi, N. Moalla, I. Walha, Spectra of some block operator matrices and application to transport operators. J. Math. Anal. Appl. 351(1), 315–325 (2009)

    MathSciNet  MATH  Google Scholar 

  176. A. Jeribi, N. Moalla, S. Yengui, S-essential spectra and application to an example of transport operators. Math. Methods Appl. Sci. 37(16), 2341–2353 (2014)

    MathSciNet  Google Scholar 

  177. A. Jeribi, S.A. Ould Ahmed Mahmoud, R. Sfaxi, Time asymptotic behavior for a one-velocity transport operator with Maxwell boundary condition. Acta Appl. Math. 3, 163–179 (2007)

    Google Scholar 

  178. Wu. Jianhong, Theory and applications of partial functional equations. Appl. Math. Sci. 119 (1996)

    Google Scholar 

  179. K. Jörgens, An asymptotic expansion in the theory of neutron transport. Commun. Pure Appl. Math. 11, 219–242 (1958)

    MATH  Google Scholar 

  180. K. Jörgens, Linear Integral Operators (Pitman Advenced Publishing Program, London, 1982)

    MATH  Google Scholar 

  181. P.E.T. Jorgensen, Essential self-adjointness of the graph-Laplacian. J. Math. Phys. 49, 073510, 33 pp. (2008)

    Google Scholar 

  182. S.J. Joseph, chemistry and algebra. Nature 17, 284 (1878). doi: 10.1038-017284a0

    Google Scholar 

  183. M.A. Kaashoek, D.C. Lay, Ascent, descent and commuting perturbations. Trans. Am. Math. Soc. 169, 35–47 (1972)

    MathSciNet  MATH  Google Scholar 

  184. H.G. Kaper, C.G. Lekkerkerker, J. Hejtmanek, Spectral Methods in Linear Transport Theory (Birkhauser, Basel, 1982)

    MATH  Google Scholar 

  185. T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators. J. Anal. Math. 6, 261–322 (1958)

    MATH  Google Scholar 

  186. T. Kato, Perturbation Theory for Linear Oerators (Springer, New York, 1966)

    Google Scholar 

  187. A. Kechris, Classical Descriptive Set Theory (Springer, New York, 1995)

    MATH  Google Scholar 

  188. M. Keller, D. Lenz, Unbounded Laplacians on graphs, Basis spectral properties and the heat equation. Math. Model. Nat. Phenom. 5(2), 27 (2009)

    Google Scholar 

  189. G. Kirchhoff, Graph theory and crystal physics, in Graph Theory and Theoretical Physics, ed. by F. Harary, Chap. 1 (Academic Press, London, 1967), pp. 44–110

    Google Scholar 

  190. D. König, Theorie der Endichen and Unendlichen Graph: Kombinatorishe Topologie der Streckenkomplexe (Akad, Leipzig, 1936)

    Google Scholar 

  191. H. Konig, Eigenvalue Distribution of Compact Operators (Birkauser, Basel, 1986)

    Google Scholar 

  192. V. Kordula, V. Müller, The distance from the Apostol spectrum. Proc. Am. Math. Soc. 124, 3055–3061 (1996)

    MATH  Google Scholar 

  193. M.A. Krasnoselskii, Positive Solutions of Operator Equations (Noordhoff, Groningen, 1964)

    Google Scholar 

  194. M.A. Krasnosel’skii, et al., Integral Operators in Space of Summabie Functions (Noordhoff, Leyden, 1976)

    Google Scholar 

  195. M.G. Krei n, M.A. Krasnoselskii, Fundamental theorems on the extension of Hermitian operators and certain of their applications to the theory of orthogonal polynomials and the problem of moments. Uspehi Matem. Nauk. 2(3(19)), 60–106 (1947)

    Google Scholar 

  196. M.G. Krein, M.A Krasnoselskii, Stability of index of an unbounded operator. Mat. Sb. (N.S.) 30(92), 219–224 (1952) (Russian)

    Google Scholar 

  197. R. Kress, Linear Integral Equations. Applied Mathematical Sciences, vol. 82 (Springer, New York, 1989)

    Google Scholar 

  198. B. Krichen, Spectral properties, fixed point theory of block operator matrices and applications to transport equations, Thesis, University of Sfax, 2011

    Google Scholar 

  199. B. Krichen, Relative essential spectra involving relative demicompact unbounded linear operators. Acta Math. Sci. 34(2), 546–556 (2014)

    MathSciNet  Google Scholar 

  200. A. Kryczka, S. Prus, Measures of weak noncompactness under complex interpolation. Stud. Math. 147, 89–102 (2000)

    MathSciNet  Google Scholar 

  201. A. Kryczka, S. Prus, M. Szczepanik, Measures of weak noncompactness and real interpolation of operators. Bull. Aust. Math. Soc. 62, 389–401 (2000)

    MathSciNet  MATH  Google Scholar 

  202. P. Kuchment, Quantum graphs, an introduction and a brief survey, ‘Analysis on graphs and its applications’, in Proc. Symp. Pure Math. (American Mathematical Society, Providence, 2008), pp. 291–314

    Google Scholar 

  203. K. Kuratowski, Sur les espaces complets. Fund. Math. 15, 301–309 (1930)

    MATH  Google Scholar 

  204. K. Kuratowski, Topology (Hafner, New York, 1966)

    Google Scholar 

  205. J-P. Labrousse, Les opérateurs quasi-Fredholm une généralisation des opérateurs semi-Fredholm. Rend. Circ. Math. Palermo 29(2), 161–258 (1980)

    MathSciNet  MATH  Google Scholar 

  206. J-P. Labrousse, Inverses généralisés d’opérateurs non bornés. Proc. Am. Math. Soc. 115(1), 125–129 (1992)

    MathSciNet  MATH  Google Scholar 

  207. V. Lakshmikantham, S. Leela, Nonlinear Differential Equations in Abstract Spaces (Pergamon Press, Oxford, 1981)

    MATH  Google Scholar 

  208. H.J. Landau, On Szegö’s eingenvalue distribution theorem and non-Hermitian kernels. J. Anal. Math. 28, 335–357 (1975)

    MATH  Google Scholar 

  209. E.W. Larsen, P.F. Zweifel, On the spectrum of the linear transport operator. J. Math. Phys. 15, 1987–1997 (1974)

    MathSciNet  Google Scholar 

  210. K. Latrach, Théorie spectrale d’équations cinétiques, Thèse, Université de Franche-Comte, 1992

    Google Scholar 

  211. K. Latrach, Compactness properties for linear transport operator with abstract boundary conditions in slab geometry. Trans. Theor. Stat. Phys. 22, 39–64 (1993)

    MathSciNet  MATH  Google Scholar 

  212. K. Latrach, Some remarks on the essential spectrum of transport operators with abstract boundary conditions. J. Math. Phys. 35(11), 6199–6212 (1994)

    MathSciNet  MATH  Google Scholar 

  213. K. Latrach, Time asymptotic behavior for linear mono-energetic transport equations with abstract boundary conditions in slab geometry. Trans. Theor. Stat. Phys. 23, 633–670 (1994)

    MathSciNet  MATH  Google Scholar 

  214. K. Latrach, Essential spectra on spaces with the Dunford-Pettis property. J. Math. Anal. Appl. 223, 607–622 (1999)

    MathSciNet  Google Scholar 

  215. K. Latrach, Compactness properties for perturbed semigroups and application to transport equation, preprint (2004)

    Google Scholar 

  216. K. Latrach, A. Dehici, Relatively strictly singular perturbations, essential spectra and application to transport operators. J. Math. Anal. Appl. 252, 767–789 (2000)

    MathSciNet  MATH  Google Scholar 

  217. K. Latrach, A. Dehici, Fredholm, semi-Fredholm perturbations and essential spectra. J. Math. Anal. Appl. 259, 277–301 (2001)

    MathSciNet  MATH  Google Scholar 

  218. K. Latrach, A. Dehici, Remarks on embeddable semigroups in groups and a generalization of some Cuthbert’s results. Int. J. Math. Math. Sci. 22, 1421–1431 (2003)

    MathSciNet  Google Scholar 

  219. K. Latrach, A. Jeribi, On the essential spectrum of transport operators on L 1-spaces. J. Math. Phys. 37(12), 6486–6494 (1996)

    MathSciNet  MATH  Google Scholar 

  220. K. Latrach, A. Jeribi, Sur une équation de transport intervenant en dynamique des populations. C. R. Acad. Sci. Paris Sér. I 325, 1087–1090 (1997)

    MathSciNet  MATH  Google Scholar 

  221. K. Latrach, A. Jeribi, Some results on Fredholm operators, essential spectra, and application. J. Math. Anal. Appl. 225, 461–485 (1998)

    MathSciNet  MATH  Google Scholar 

  222. K. Latrach, B. Lods, Regularity and time asymptotic behavior of solutions to transport equations. Trans. Theor Stat. Phys. 30, 617–639 (2001)

    MathSciNet  MATH  Google Scholar 

  223. K. Latrach, H. Megdiche, A. Jeribi, Time asymptotic behavior of the solution to a Cauchy problem governed by a transport operator. J. Intergr. Equ. Appl. 17(2), 121–140 (2005)

    MathSciNet  MATH  Google Scholar 

  224. K. Latrach, J.M. Paoli, Relatively compact-like perturbations, essentilal spectra and applications. J. Aust. Math. Soc. 77(1), 73–89 (2004)

    MathSciNet  MATH  Google Scholar 

  225. K. Latrach, J.M. Paoli, Polynomially compact-like strongly continuous semigroups. Acta Appl. Math. 82, 87–99 (2004)

    MathSciNet  MATH  Google Scholar 

  226. K. Latrach, J.M. Paoli, An extension of a Phillips’s theorem to Banach algebras and application to the uniform continuity of strongly continuous semigroups. J. Math. Anal. Appl. 326, 945–959 (2007)

    MathSciNet  MATH  Google Scholar 

  227. K. Latrach, J.M. Paoli, P. Simonnet, Some facts from descriptive set theory concerning essential spectra and applications. Stud. Math. 171, 207–225 (2005)

    MathSciNet  MATH  Google Scholar 

  228. K. Latrach, J.M. Paoli, P. Simonnet, A spectral characterization of the uniform continuity of strongly continuous groups. Arch. Math. 90, 420–428 (2008)

    MathSciNet  MATH  Google Scholar 

  229. K. Latrach, J.M. Paoli, M.A. Taoudi, A characterization of polynomially Riesz strongly continuous semigroups. Comment. Math. Univ. Carol. 47(2), 275–289 (2006)

    MathSciNet  MATH  Google Scholar 

  230. A. Lebow, M. Schechter, Semigroups of operators and measures of noncompactness. J. Funct. Anal. 7, 1–26 (1971)

    MathSciNet  MATH  Google Scholar 

  231. J. Lehner, M. Wing, On the spectrum of an unsymetric operator arisingin the transport theory of neutrons. Commun. Pure Appl. Math. 8, 217–234 (1955)

    MathSciNet  MATH  Google Scholar 

  232. J. Lehner, M. Wing, Solution of the linearized Boltzmann transport equation for the slab geometry. Duke Math. 23, 125–142 (1956)

    MathSciNet  MATH  Google Scholar 

  233. A.E. Lifschitz, Magnetohydrodynamics and Spectral Theory (Springer, Dordrecht, 1989)

    MATH  Google Scholar 

  234. J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. 1 (Springer, Berlin-Heidelberg, 1972)

    Google Scholar 

  235. C. Lizama, Uniform continuity and compactness for resolvent families of operators. Acta Appl. Math. 38, 131–138 (1995)

    MathSciNet  MATH  Google Scholar 

  236. B. Lods, On linear kinetic equations involving unbounded cross-sections. Math. Models Methods Appl. Sci. 27, 1049–1075 (2004)

    MathSciNet  MATH  Google Scholar 

  237. H.P. Lotz, Über das Spektrum positiver Operatoren. Math. Z. 108, 15–32 (1968)

    MathSciNet  MATH  Google Scholar 

  238. A. Lunardi, Analytic Semigroups and Optimal Regularity in Pàarabolic Problems (Birkhauser, Basel, 1995)

    Google Scholar 

  239. J. Lutgen, On essential spectra of operator-matrices and their Feshbach maps. J. Math. Anal. Appl. 289, 419–430 (2004)

    MathSciNet  MATH  Google Scholar 

  240. D. Lutz, Compactness properties of operator cosine functions. C. R. Math. Rep. Acad. Sci. Can. 2, 277–280 (1980)

    MathSciNet  MATH  Google Scholar 

  241. I. Marek, Frobenius theory of positive operators: Comparison theorems and applications. SIAM J. Appl. Math. 19, 607–628 (1970)

    MathSciNet  MATH  Google Scholar 

  242. I. Marek, Fundamental decay and asymptotic behavior of positive semig roups. Czechoslov. Math. J. 30(105), 579–590 (1980)

    MathSciNet  Google Scholar 

  243. A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils (American Mathematical Society, Providence, 1988)

    MATH  Google Scholar 

  244. J.E. Marsden, Basic Complex Analysis (W. H. Freeman and Campany, San Francisco, 1973)

    MATH  Google Scholar 

  245. A. Mashaghi, Inverstigation of a protein complex network. Eur. Phys. 41(1), 113–121 (2004)

    Google Scholar 

  246. M. Mbekhta, A. Ouahab, Opérateur s-régulier dans un espace de Banach et théorie spectrale. Acta Sci. Math. 59, 525–543 (1994)

    MathSciNet  MATH  Google Scholar 

  247. V. Menon, On repeated interchange graphs. Am. Math. Mon. 13, 986–989 (1966)

    MathSciNet  Google Scholar 

  248. R. Mennicken, S. Naboko, C. Tretter, Essential spectrum of a system singular differential operators and the asymptotic Hain-Lüst operator. Am. Math. Soc. 130, 1699–1710, (2001)

    MathSciNet  Google Scholar 

  249. P. Meyer-Nieberg, Banach Lattices (Springer, New York, 1991)

    MATH  Google Scholar 

  250. O. Milatovic, Essential self-adjointness of magnetic Schrödinger operators on locally finite graph. Integr. Equ. Oper. Theory 71, 13–27 (2011)

    MathSciNet  MATH  Google Scholar 

  251. O. Milatovic, A sears-type self-adjointness result for discrete magnetic Schrödinger operators. J. Math. Anal. Appl. 369, 801–809 (2012)

    MathSciNet  Google Scholar 

  252. V.D. Milman, Some properties of strictly singular operators. Funct. Anal. Appl. 3, 77–78 (1969)

    Google Scholar 

  253. M.M. Milovanović-Arandjelović, Measures of noncompactness on uniform spaces- the axiomatic approach, in IMC “Filomat 2001”, Niš (2001), pp. 221–225

    Google Scholar 

  254. N. Moalla, Developpement de certaines propriétés fines de la théorie spectrale et applications à l’équation de transport, Thesis, University of Sfax, 2006

    Google Scholar 

  255. N. Moalla, A characterization of Schechter’s essential spectra by mean of measure of non-strict-singularity and application to matrix operator. Acta Math. Sci. Ser. B Engl. Ed. 32(6), 2329–2340 (2012)

    MathSciNet  MATH  Google Scholar 

  256. N. Moalla, M. Damak, A. Jeribi, Essential spectra of some matrix operators and application to two-group transport operators with general boundary conditions. J. Math. Anal. Appl. 323(2), 1071–1090 (2006)

    MathSciNet  MATH  Google Scholar 

  257. B. Mohar, W. Woess, A survey on spectra of infinite graphs. J. Bull. Lond. Math. Soc. 21(3), 209–234 (1989)

    MathSciNet  MATH  Google Scholar 

  258. M. Mokhtar-Kharroubi, Propriétés spéctrales de l’opérateur de transport dans le cas anisotrope, Thèse de Doctorat de 3ème cycle, Université Paris 6, 1983

    Google Scholar 

  259. M. Mokhtar-Kharroubi, Quelques applications de la positivité en théorie du transport. Ann. Fac. Sci. Toulouse. 11, 75–99 (1990)

    Google Scholar 

  260. M. Mokhtar-Kharroubi, Compactness results for positive semigroups on Banach Lattices and applications. Houston J. Math. 17(1), 25–38 (1991)

    MathSciNet  MATH  Google Scholar 

  261. M. Mokhtar-Kharroubi, Time asymptotic bahaviour and compactness in Neutron Transport Theory. Eur. J. Mech. B Fluids 11(1), 39–68 (1992)

    MathSciNet  Google Scholar 

  262. M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. New Aspects, vol. 46 (World Scientific, Singapore, 1997)

    Google Scholar 

  263. V. Müller, On the regular spectrum. J. Oper. Theory 31, 363–80 (1994)

    MATH  Google Scholar 

  264. V. Müller, Spectral theory of linear operators and spectral system in Banach algebras. Oper. Theor. Adv. Appl. 139 (2003)

    Google Scholar 

  265. R. Nagel, Towards a “matrix theory” for unbounded operator matrices. Math. Z. 201(1), 57–68 (1989)

    MathSciNet  MATH  Google Scholar 

  266. R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain. J. Funct. Anal. 89(2), 291–302 (1990)

    MathSciNet  MATH  Google Scholar 

  267. B.Sz. Nagy, On the stability of the index of unbounded linear transformations. Acta. Math. Acad. Sic. Hungar. 3, 49–52 (1952)

    MATH  Google Scholar 

  268. M.A. Naimark, Linear Differential Operators (Frederick Ungar, New York, 1987)

    Google Scholar 

  269. L.I. Nicolaescu, On the space of Fredholm operators. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 53(2), 209–227 (2007)

    Google Scholar 

  270. J.I. Nieto, On Fredholm operators and the essential spectrum of singular integral operators. Math. Ann. 178, 62–77 (1968)

    MathSciNet  MATH  Google Scholar 

  271. R.D. Nussbaum, Positive operators and elliptic eigenvalue problems. Math. Z. 186, 247–264 (1984)

    MathSciNet  MATH  Google Scholar 

  272. W. Obershelp, Theory of Graphs, vol. 38 (American Mathematical Society Colloquium Publications, Providence, 1963)

    Google Scholar 

  273. Z. Opial, Nonexpansive and Monotone Mappings in Banach Spaces (Center for Dynamical Systems, Brown University, Providence, 1967), pp. 1–67

    Google Scholar 

  274. A. Palczewski, Spectral properties of the space nonhomogeneous linearized-Boltzmann operator. ’Ikansp. Theor. Stat. Phys. 13, 409–430 (1984)

    Google Scholar 

  275. C.V. Pao, Asymptotic behavior of the solution for the time-dependent neutron transport problem. J. Integr. Equ. 1, 31–152 (1979)

    MathSciNet  Google Scholar 

  276. A. Pazy, Semigroups of Linear Operators and Applications to Differential Equations. Applied Mathematical Sciences, vol. 44 (Springer, New York, 1983)

    Google Scholar 

  277. A. Pelczynski, On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in C(X)-spaces. II. Strictly singular and strictly cosingular operators in L(μ)-spaces. Bull. Acad. Polon. Sci. 13, 13–36, 37–41 (1965)

    Google Scholar 

  278. S. Pemmaraju, S. Skiena, Cycles, stars, and wheels, in Computational Discrete Mathematics Combinatiorics and Graph Theory in Mathematica, section 6.4 (Cambridge University Press, Cambridge, 2003), pp. 284–249

    Google Scholar 

  279. W.V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space. J. Math. Anal. Appl. 14, 276–284 (1966)

    MathSciNet  MATH  Google Scholar 

  280. W.V. Petryshyn, Remarks on condensing and k-set-contractive mappings. J. Math. Anal. Appl. 39, 717–741 (1972)

    MathSciNet  MATH  Google Scholar 

  281. R.S. Phillips, Spectral theory for semigroups of linear operators. Trans. Am. Math. Soc. 71, 393–415 (1951)

    MATH  Google Scholar 

  282. O. Post, First order approach and index theorems for discrete and metric graph. Am. Heni. Poincaré 10, 823–866 (2009)

    MathSciNet  MATH  Google Scholar 

  283. F. Rabiger, W.J. Ricker, C 0-groups and C 0-semigroups of linear operators on hereditarily indecomposable Banach spaces. Arch. Math. 66, 60–70 (1996)

    MathSciNet  Google Scholar 

  284. V. Rakoćević, On one subset of M. Schechter’s essential spectrum. Mat. Vesnik 5(18)(33)(4), 389–391 (1981)

    Google Scholar 

  285. V. Rakoc̆ević, Approximate point spectrum and commuting compact perturbation. Glasgow Math. J. 28, 193–198 (1986)

    Google Scholar 

  286. V. Rakocevic̀, Generalized spectrum and commuting compact perturbations. Proc. Edinb. Math. Soc. 36, 197–209 (1993)

    Google Scholar 

  287. V. Rako\(\check{\text{c}}\) evi\(\acute{\text{c}}\), Semi-Fredholm operators with finite ascent or descent and perturbations. Am. Math. Soc. 123(12) (1995)

    Google Scholar 

  288. V. Rakoc̃ević, Semi-Browder operators and perturbations. Stud. Math. 122(2), 131–137 (1997)

    Google Scholar 

  289. V. Rakočević, Measures of noncompactness and some applications. Filo-Mat. 12(2), 87–120 (1998)

    MATH  Google Scholar 

  290. J.S. Raymond, Boréliens à coupes K σ . Bull. Soc. Math. France 104, 389–406 (1976)

    MathSciNet  MATH  Google Scholar 

  291. M. Reed, Linear graphs and Electrical Networks (Addisson Wesky, Reading, 1961)

    MATH  Google Scholar 

  292. M. Reed, B. Simon, Methods of Modern Mathematical Physics, I-IV. Analysis of Operators (Academic Press, New York, 1978)

    Google Scholar 

  293. M. Ribaric, I. Vidav, Analytic Properties of the Inverse A(z) −1 of an Analytic Linear Operator Valued Function A(z). Arch. Ration. Mech. Anal. 32(4), 298–310 (1969)

    MathSciNet  MATH  Google Scholar 

  294. F. Riesz, Über lineare funktionalgleichungen. Acta Math. 41, 71–98 (1918)

    MathSciNet  Google Scholar 

  295. N. Robertson, D. Sanders, P. Seymour, R. Thomas, The four color theorem. J. Comb. Theory Ser. B 70, 2–44 (1997)

    MathSciNet  MATH  Google Scholar 

  296. M. Rotenberg, Transport theory for growing cell populations. J. Theor. Biol. 103, 181–199 (1983)

    MathSciNet  Google Scholar 

  297. H. Sachs, Graph derivatives. Math. Z. 76, 385–401 (1961)

    MathSciNet  Google Scholar 

  298. H.H. Schaefer, Banach lattices and positive operators. Grundlehren Math. Wiss. Bd., vol. 215 (Springer, New York, 1974)

    Google Scholar 

  299. M. Schechter, On the essential spectrum of an arbitrary operator. J. Math. Anal. Appl. 13, 205–215 (1966)

    MathSciNet  MATH  Google Scholar 

  300. M. Schechter, Basic theory of Fredholm operators. Anna. Scuola Norm. Sup. Pisa 21(3), 261–280 (1967)

    MathSciNet  MATH  Google Scholar 

  301. M. Schechter, Spectra of Partial Differential Operators (North-Holland, Amsterdam, 1971)

    MATH  Google Scholar 

  302. M. Schechter, Principles of Functional Analysis. Graduate Studies in Mathematics, vol. 36 (American Mathematical Society, Providence, 2002)

    Google Scholar 

  303. C. Schmoeger, Perturbation properties of some class of operators. Rend. Math. Appl. 7, 533–541 (1994)

    MathSciNet  Google Scholar 

  304. C. Schmoeger, The spectral mapping theorem for the essential approximate point spectrum. Colloq. Math. 74(2), 167–176 (1997)

    MathSciNet  MATH  Google Scholar 

  305. I. Schur, Bemerkungen Zur theorie der Beschrankten Bilinear formen mit unendhich vielen Veranderhichen, J. Reine Angew. Math. 140 1–28 (1911)

    MathSciNet  MATH  Google Scholar 

  306. G.P. Shannoa, Strictly singular and cosingular operators and topological vector spaces. Proc. R. lr. Acad. Sect. A 73, 303–308 (1973)

    Google Scholar 

  307. J. Shapiro, M. Schechter, A generalized operational calculus developed from Frdholm operator theory. Trans. Am. Math. Soc. 175, 439–667 (1973)

    MathSciNet  MATH  Google Scholar 

  308. J. Shapiro, M. Snow, The Fredholm spectrum of the sum and product of two operators. Trans. Am. Math. Soc. 191, 387–393 (1974)

    MathSciNet  MATH  Google Scholar 

  309. A.A. Shkalikov, On the essential spectrum of some matrix operators. Math. Notes 58(5–6), 1359–1362 (1995)

    MathSciNet  MATH  Google Scholar 

  310. A.A. Shkalikov, C. Tretter, Spectral analysis for linear pencils Nλ P of ordinary differential operators. Math. Nachr. 179, 275–305 (1996)

    MathSciNet  MATH  Google Scholar 

  311. Yu.L. Smul’Yan, Completely continuous perturbation of operators. Dokl. Akad. Nauk SSSR (N.S.) 101, 35–38 (1955) (Russian)

    Google Scholar 

  312. D. Song, Some notes on the spectral properties of C 0-semigroups generated by linear transport operators. Trans. Theor. Stat. Phys. 26, 233–242 (1997)

    MATH  Google Scholar 

  313. D. Song, On the spectrum of neutron transport equations with reflecting boundary conditions, PhD Thesis, Blacksburg, 2000

    Google Scholar 

  314. S. Steinberg, Meromorphic families of compact operators. Arch. Rational Mech. Anal. 31, 372–379 (1968)

    MathSciNet  MATH  Google Scholar 

  315. K. Taira, A. Favini, S. Romanelli, Feller semigroups and degenerate elliptic operators with Wentzell boundary conditions. Stud. Math. 145, 17–53 (2001)

    MathSciNet  MATH  Google Scholar 

  316. P. Takac, A spectral mapping theorem for the exponential function in linear transport theory. Trans. Theor. Stat. Phys. 14, 655–667 (1985)

    MathSciNet  MATH  Google Scholar 

  317. W.T. Tatte, Graph Theory (Cambridge university Press, Cambridge, 2001), p. 30

    Google Scholar 

  318. A.E. Taylor, Spectral theory of closed distributive operators. Acta Math. 84, 189–224, MR 12, 717 (1951)

    Google Scholar 

  319. A.E. Taylor, Theorems on ascent, descent, nullity, and defect of linear operators. Math. Ann. 163, 18–49 (1966)

    MathSciNet  MATH  Google Scholar 

  320. M. Taylor, Partial Differential Equations. Basic Theory, vol. 1 (Springer, New York, 1996)

    Google Scholar 

  321. T. Toka, Perturbations of Unbounded Fredholm Linear Operators in Banach Spaces, Handbook on Operator Theory (Springer, 2015)

    Google Scholar 

  322. L.N. Trefethen, Pseudo-spectra of matrices, in Numer. Anal. 1991 (Longman Scientific & Technical, Harlow, 1992), pp. 234–266

    Google Scholar 

  323. C. Tretter, Spectral issues for block operator matrices, in Differential Equations and Mathematical Physics, Birmingham, 1999; AMS/IP Studies in Advanced Mathematics, vol. 16 (American Mathematical Society, Providence, 2000), pp. 407–423

    Google Scholar 

  324. C. Tretter, Spectral Theory of Block Operator Matrices and Applications (Impe. Coll. Press, London, 2008)

    MATH  Google Scholar 

  325. C. Tretter, Spectral inclusion for unbounded block operator matrices. J. Funct. Anal. 11, 3806–3829 (2009)

    MathSciNet  Google Scholar 

  326. R. Van Norton, On the real spectrum of a monoenergetic neutron transport operator. Commun. Pure Appl. Math. 15, 149–158 (1962)

    MATH  Google Scholar 

  327. J.M. Varah, The Computation of Bounds for the Invariant Subspaces of a General Matrix Operator, Stan. Univ. Comp. Sci., Dept. Tech. Report (1967)

    Google Scholar 

  328. I. Vidav, Existence and uniqueness of nonnegative eigenfunction of the Boltzmann operator. J. Math. Anal. Appl. 22, 144–155 (1968)

    MathSciNet  MATH  Google Scholar 

  329. I. Vidav, Spectra of perturbed semigroups with applications to transport - theory. J. Math. Anal. Appl. 30, 264–279 (1970)

    MathSciNet  MATH  Google Scholar 

  330. Ju.I. Vladimirskii, Stricty cosingular operators. Sov. Math. Dokl. 8, 739–740 (1967)

    Google Scholar 

  331. J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups. Mh. Math. 90, 153–161 (1980)

    MathSciNet  MATH  Google Scholar 

  332. J. Voigt, Functional Analytic Treatment of the Initial Boundary Value Problem for Collisionlgs Gases (Habilitationsschrift, Munchen, 1981)

    Google Scholar 

  333. J. Voigt, Spectral properties of the neutron transport equation. J. Math. Anal. Appl. 106, 140–153 (1985)

    MathSciNet  MATH  Google Scholar 

  334. J. Voigt, On resolvent positive operators and positive C 0-semigroup on AL-spaces. Semigroup Forum 38, 263–266 (1989)

    MathSciNet  MATH  Google Scholar 

  335. I. Walha, Essential spectra of some operator matrices, Riesz basis and applications, Thesis, University of Sfax, 2010

    Google Scholar 

  336. G.F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (Marcel Dekker, New York, 1985)

    MATH  Google Scholar 

  337. A. Weber, Analysis of the physical Laplacian and the heat flow on a locally finite graphs. J. Math. Anal. 370, 146–158 (2010)

    MATH  Google Scholar 

  338. L. Weis, Perturbation class of semi-Fredholm operators. Math. Z. 178, 429–442 (1981)

    MathSciNet  MATH  Google Scholar 

  339. L.W. Weis, A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to transport theory. J. Math. Anal. Appl. 129, 6–23 (1988)

    MathSciNet  MATH  Google Scholar 

  340. T.T. West, Riesz operators in Banach spaces. Proc. Lond. Math. Soc. 16, 131–140 (1966)

    MATH  Google Scholar 

  341. T.T. West, A Riesz-Schauder theorem for semi-Fredholm operators. Proc. R. Ir. Acad. Sect. A 87, 137–146 (1987)

    MATH  Google Scholar 

  342. H. Weyl, Uber beschrankte quadratiche Formen, deren Differenz vollsteig ist. Rend. Circ. Mat. Palermo 27, 373–392 (1909)

    MATH  Google Scholar 

  343. R.J. Whitley, Strictly singular operators and their congugates. Trans. Am. Math. Soc. 18, 252–261 (1964)

    MathSciNet  Google Scholar 

  344. M. Wing, An Introduction to Transport Theory (Wiley, New York, 1962)

    Google Scholar 

  345. R. Wojciechowski, Stochastic compactetness of graph, Ph.D. thesis, City University of New York, 72 pp., 2007

    Google Scholar 

  346. R. Wojciechowski, Stochatically incomplete manifolds and graphs. Progr. Probab. 64, 163–179 (2011)

    MathSciNet  Google Scholar 

  347. F. Wolf, On the invariance of the essential spectrum under a change of the boundary conditions of partial differential operators. Indag. Math. 21, 142–147 (1959)

    Google Scholar 

  348. F. Wolf, On the essential spectrum of partial differential boundary problems. Commun. Pure Appl. Math. 12, 211–228 (1959)

    MATH  Google Scholar 

  349. M.P.H. Wolff, Discrete approximation of unbounded operators and approximation of their spectra. J. Approx. Theory 113, 229–244 (2001)

    MathSciNet  MATH  Google Scholar 

  350. Z. Xianwen, Spectral properties of a streaming operator with diffuse reflection boundary condition. J. Math. Anal. Appl. 238, 20–43 (1999)

    MathSciNet  MATH  Google Scholar 

  351. M. Yahdi, Théorie descriptive des ensembles en géométrie des espaces de Banach, exemples, Thése de Doctorat de Mathématiques, Université Paris 6, 1998

    Google Scholar 

  352. S. Yengui, S-spectres essentiels, theorie de perturbation et applications à l’équation de transport, Thesis, University of Sfax, 2012

    Google Scholar 

  353. B. Yood, Properties of linear transformations preserved under addition of a completely continuous transformation. Duke Math. J. 18, 599–612 (1951)

    MathSciNet  MATH  Google Scholar 

  354. K. Yosida, Functional Analysis (Springer, Heidelberg, 1978)

    MATH  Google Scholar 

  355. A.C. Zaanen, Riesz Spaces II (North Holland, Amsterdam, 1983)

    MATH  Google Scholar 

  356. M. Zerner, Quelques propriétés spectrales des opérateurs positifs. J. Funct. Anal. 72, 381–417 (1987)

    MathSciNet  MATH  Google Scholar 

  357. X. Zhang, B. Liang, On the spectum of a one-velocity transport operator with Maxwell boundary condition. J. Math. Anal. Appl. 202, 920–936 (1996)

    MathSciNet  MATH  Google Scholar 

  358. S. \(\check{\text{Z}}\) ivkovi\(\acute{\text{c}}\), Semi-Fredholm operators and perturbation. Publ. Inst. Math. Beo. 61, 73–89 (1997)

    Google Scholar 

  359. S. \(\check{\text{Z}}\) ivkovi\(\acute{\text{c}}\)-Zlatanovi\(\grave{\text{c}}\), D.S. Djordjevi\(\acute{\text{c}}\), R.E. Harte, On left and right Browder operators. J. Kor. Math. Soc. 485, 1053–1063 (2011)

    Google Scholar 

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Jeribi, A. (2015). Fredholm Theory Related to Some Measures. In: Spectral Theory and Applications of Linear Operators and Block Operator Matrices. Springer, Cham. https://doi.org/10.1007/978-3-319-17566-9_5

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