Ren, Guojing; Sun, Huaqing \(J\)-self-adjoint extensions for a class of discrete linear Hamiltonian systems. (English) Zbl 1470.47019 Abstr. Appl. Anal. 2013, Article ID 904976, 19 p. (2013). Summary: This paper is concerned with formally \(J\)-self-adjoint discrete linear Hamiltonian systems on finite or infinite intervals. The minimal and maximal subspaces are characterized, and the defect indices of the minimal subspaces are discussed. All the \(J\)-self-adjoint subspace extensions of the minimal subspace are completely characterized in terms of the square summable solutions and boundary conditions. As a consequence, characterizations of all the \(J\)-self-adjoint subspace extensions are given in the limit point and limit circle cases. Cited in 2 Documents MSC: 47B25 Linear symmetric and selfadjoint operators (unbounded) × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Akhiezer, N. I.; Glazman, I. M., Theory of Linear Operators in Hilbert Space (1981), London, UK: Pitman Publishing, London, UK · Zbl 0467.47001 [2] Dunford, N.; Schwartz, J. T., Linear Operators. Part II: Spectral theory. Self Adjoint Operators in Hilbert Space (1963), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0128.34803 [3] Glazman, I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (1965), Jerusalem, Israel: Israel Program for Scientific Translations, Jerusalem, Israel · Zbl 0143.36505 [4] Weidmann, J., Spectral Theory of Ordinary Differential Operators, 1258 (1987), Berlin, Germany: Springer, Berlin, Germany · Zbl 0647.47052 [5] Sun, H.; Shi, Y., Self-adjoint extensions for singular linear Hamiltonian systems, Mathematische Nachrichten, 284, 5-6, 797-814 (2011) · Zbl 1228.47026 · doi:10.1002/mana.200810235 [6] Sun, H.; Shi, Y., Self-adjoint extensions for linear Hamiltonian systems with two singular endpoints, Journal of Functional Analysis, 259, 8, 2003-2027 (2010) · Zbl 1202.47012 · doi:10.1016/j.jfa.2010.06.008 [7] Sun, J., On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices, Acta Mathematica Sinica, 2, 2, 152-167 (1986) · Zbl 0615.34015 · doi:10.1007/BF02564877 [8] Wang, A.; Sun, J.; Zettl, A., Characterization of domains of self-adjoint ordinary differential operators, Journal of Differential Equations, 246, 4, 1600-1622 (2009) · Zbl 1169.47033 · doi:10.1016/j.jde.2008.11.001 [9] Shi, Y., The Glazman-Krein-Naimark theory for Hermitian subspaces, Journal of Operator Theory, 68, 1, 241-256 (2012) · Zbl 1260.47007 [10] Shi, Y., Weyl-Titchmarsh theory for a class of discrete linear Hamiltonian systems, Linear Algebra and Its Applications, 416, 2-3, 452-519 (2006) · Zbl 1100.39020 · doi:10.1016/j.laa.2005.11.025 [11] Coddington, E. A., Extension Theory of Formally Normal and Symmetric Subspaces, 134 (1973), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0265.47023 [12] Coddington, E. A., Self-adjoint subspace extensions of nondensely defined symmetric operators, Bulletin of the American Mathematical Society, 79, 712-715 (1973) · Zbl 0285.47020 · doi:10.1090/S0002-9904-1973-13275-6 [13] Coddington, E. A.; Dijksma, A., Self-adjoint subspaces and eigenfunction expansions for ordinary differential subspaces, Journal of Differential Equations, 20, 2, 473-526 (1976) · Zbl 0306.34023 · doi:10.1016/0022-0396(76)90119-4 [14] Dijksma, A.; de Snoo, H. S. V., Self-adjoint extensions of symmetric subspaces, Pacific Journal of Mathematics, 54, 71-100 (1974) · Zbl 0304.47006 · doi:10.2140/pjm.1974.54.71 [15] Lesch, M.; Malamud, M., On the deficiency indices and self-adjointness of symmetric Hamiltonian systems, Journal of Differential Equations, 189, 2, 556-615 (2003) · Zbl 1016.37026 · doi:10.1016/S0022-0396(02)00099-2 [16] Ren, G.; Shi, Y., Self-adjoint extensions for discrete linear Hamiltonian systems · Zbl 1291.39018 [17] Shi, Y.; Sun, H., Self-adjoint extensions for second-order symmetric linear difference equations, Linear Algebra and its Applications, 434, 4, 903-930 (2011) · Zbl 1210.39004 · doi:10.1016/j.laa.2010.10.003 [18] Glazman, I. M., An analogue of the extension theory of Hermitian operators and a non-symmetric one-dimensional boundary problem on a half-axis, Doklady Akademii Nauk SSSR, 115, 214-216 (1957) · Zbl 0079.33102 [19] Race, D., The theory of \(J\)-self-adjoint extensions of \(J\)-symmetric operators, Journal of Differential Equations, 57, 258-274 (1985) · Zbl 0525.47016 [20] Shang, Z., On \(J\)-self-adjoint extensions of \(J\)-symmetric ordinary differential operators, Journal of Differential Equations, 73, 153-177 (1988) · Zbl 0664.34037 [21] Monaquel, S. J.; Schmidt, K. M., On \(M\)-functions and operator theory for non-self-adjoint discrete Hamiltonian systems, Journal of Computational and Applied Mathematics, 208, 1, 82-101 (2007) · Zbl 1127.39045 · doi:10.1016/j.cam.2006.10.043 [22] Sun, H.; Qi, J., The theory for \(J\)-Hermitian subspaces in a product space, ISRN Mathematical Analysis, 2012 (2012) · Zbl 1264.47007 · doi:10.5402/2012/676835 [23] Ren, G.; Shi, Y., Defect indices and definiteness conditions for a class of discrete linear Hamiltonian systems, Applied Mathematics and Computation, 218, 7, 3414-3429 (2011) · Zbl 1256.39004 · doi:10.1016/j.amc.2011.08.086 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. 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