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Articles

Boundedness and closedness of linear relations

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Pages 309-333 | Received 22 Aug 2016, Accepted 17 Feb 2017, Published online: 09 Mar 2017

References

  • Dereziński J. Unbounded linear operators. Warszawa: Lect. Notes; 2013.
  • Kato T. Perturbation theory for linear operators. 2nd ed. Berlin: Springer-Verlag; 1984.
  • Reed M, Simon B. Methods of modern mathematical physics II: Fourier analysis, self-adjointness. New York (NY): Academic Press; 1972.
  • Schmüdgen K. Unbounded self-adjoint operators on Hilbert space. Dordrecht: Springer; 2012.
  • Weidmann J. Linear operators in Hilbert spaces. New York (NY): Springer-Verlag; 1980. (Graduate texts in mathematics; vol. 68).
  • Ren G, Shi Y. Defect indices and definiteness conditions for discrete linear Hamiltonian systems. Appl Math Comput. 2011;218:3414–3429.
  • Shi Y, Sun H. Self-adjoint extensions for second-order symmetric linear difference equations. Linear Algebra Appl. 2011;434:903–930.
  • Lesch M, Malamud M. On the deficiency indices and self-adjointness of symmetric Hamiltonian systems. J Differ Equ. 2003;18:556–615.
  • Coddington EA. Extension theory of formally normal and symmetric subspaces. Mem Am Math Soc. 1973;134.
  • Von Neumann J. Über adjungierte funktional-operatoren. Ann Math. 1932;33:294–340.
  • Von Neumann J. Functional operators II: the geometry of orthogonal spaces. Princeton (NJ): Princeton University Press; 1950. (Annals of mathematics studies; vol. 22).
  • Arens R. Operational calculus of linear relations. Pac J Math. 1961;11:9–23.
  • Álvarez T. Small perturbation of normally solvable relations. Publ Math Debrecen. 2012;80(1–2):155–68.
  • Álvarez T, Ammar A, Jeribi A. On the essential spectra of some matrix of linear relations. Math Meth Appl Sci. 2014;37:620–644.
  • Ya Azizov T, Behrndt J, Jonas P, et al. Compact and finite rank perturbations of linear relations in Hilbert spaces. Integr Equ Oper Theory. 2009;63:151–163.
  • Ya Azizov T, Behrndt J, Jonas P, et al. Spectral points of definite type and type π for linear operators and relations in Krein spaces. J Lond Math Soc. 2011;83:768–788.
  • Chafai E, Mnif M. Perturbation of normally solvable linear relations in paracomplete spaces. Linear Algebra Appl. 2013;439:1875–1885.
  • Coddington EA, Dijksma A. Adjoint subspaces in Banach spaces with applications to ordinary differential subspaces. Ann Mat Pura Appl. 1978;118:1–118.
  • Wilcox D. Multivalued semi-Fredholm operators in normed linear spaces. [PhD thesis]. Cape Town: University of Cape Town; 2002.
  • Cross R. Multivalued linear operators. New York (NY): Marcel Dekker; 1998. (Monographs and textbooks in pure and applied mathematics, vol. 213).
  • Ammar A, Diagana T, Jeribi A. Perturbations of Fredholm linear relations in Banach spaces with application to 300D73-block matrices of linear relations. Arab J Math Sci. 2015;22(1):59–76.
  • Ammar A, Fakhfakh S, Jeribi A. Stability of the essential spectrum of the diagonally and off-diagonally dominant block matrix linear relations. J Pseudo-Differ Oper Appl. 2016;17 p. DOI: 10.1007/s11868-016-0154-z
  • Hassi S, De Snoo H. One-dimensional graph perturbations of self-adjoint relations. Ann Acad Sci Fenn Math. 1997;20:123–164.
  • Hassi S, De Snoo H, Szafraniec FH. Componentwise and cartesian decompositions of linear relations. Dissertationes Math. 2009;465:59 p.
  • Lee SJ, Nashed MZ. Algebraic and topological selections of multi-valued linear relations. Ann Scuola Norm Sup Piss. 1990;17:111–126.
  • Lee SJ, Nashed MZ. Normed linear relations: domain decomposability, adjoint subspaces and selections. Linear Algebra Appl. 1991;153:135–159.
  • Shi Y. The Glazman-Krein-Naimark theory for Hermitian subspaces. J Oper Theor. 2012;68(1):241–256.
  • Shi Y. Stability of essential spectra of self-adjoint subspaces under compact perturbations. J Math Anal Appl. 2016;433:832–851.
  • Shi Y, Shao C, Liu Y. Resolvent convergence and spectral approximations of sequences of self-adjoint subspaces. J Math Anal Appl. 2014;409:1005–1020.
  • Shi Y, Shao C, Ren G. Spectral properties of self-adjoint subspaces. Linear Algebra Appl. 2013;438:191–218.
  • Wilcox D. Essential spectra of linear relations. Linear Algebra Appl. 2014;462:110–125.
  • Lee SJ, Nashed MZ. Least-squares solutions of multi-valued linear operator equations in Hilbert spaces. J Approx Theory. 1983;38:380–391.
  • Lee SJ, Nashed MZ. Constrained least-squares solutions of linear inclusions and singular control problems in Hilbert spaces. Appl Math Optim. 1989;19:225–242.
  • Liu Y, Shi Y. Regular approximations of spectra of singular second-order symmetric linear difference equations. Linear Algebra Appl. 2014;444:183–210.
  • Luenberger DG. Optimization by vector space methods. New York (NY): Wiley; 1969.
  • Nashed MZ. Operator parts and generalized inverses of linear manifolds with applications. In: Lakshmikantham V, editor. Trends in theory and practice of nonlinear differential equations. New York (NY): Marcel Dekker; 1984. p. 395–412.
  • Ren G, Shi Y. Self-adjoint extensions of for a class of discrete linear Hamiltonian systems. Linear Algebra Appl. 2014;454:1–48.
  • Robinson SM. Normed convex processes. Trans Am Math Soc. 1972;174:124–140.
  • Sun H, Shi Y. Spectral properties of singular discrete linear Hamiltonian systems. J Differ Equ Appl. 2014;20:379–405.

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