Complex symplectic geometry with applications to vector differential operators

Let l(y) be a formally self-adjoint vector-valued differential expression of order n on an interval (a, \infty)(-\infty \leq a < \infty) with complex matrix-valued function coefficients and finite equal deficiency indices. In this paper, applying complex symplectic algebra, we give a reformulation for self-adjoint domains of the minimal operator associated with l(y) and classify them.

Anahtar Kelimeler:

Symplectic algebra, Lagrangian subspace, vector-valued differential operator, self-adjoint domains

Complex symplectic geometry with applications to vector differential operators

Keywords:

Symplectic algebra, Lagrangian subspace, vector-valued differential operator, self-adjoint domains,

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Coddington, E. A.: The spectral representation of ordinary self-adjoint diﬀerential operators. Ann. Math. 60, 192–211 (1954).
Dunford, N., Schwartz, J. T.: Linear operators II. New York, Wiley 1963.
Everitt, W. N., Markus, L.: Boundary value problem and symplectic algebra for ordinary diﬀerential and quasidiﬀerential operators. Math. Surveys and Monographs 61, Amer. Math. Soc. 1999.
Everitt, W. N., Markus, L.: Complex symplectic geometry with applications to ordinary diﬀerential operators. Trans. Amer. Math. Soc. 351, 4905–4945 (1999).
Everitt, W. N., Markus, L.: Complex symplectic spaces and boundary value problems. Bulletin Amer. Math. Soc. (New Series) 42, 461–500 (2005).
Fu, S. Z.: On the self-adjoint extensions of symmetric ordinary diﬀerential operators in direct sum spaces. J. Diﬀ. Equa. 100, 269–291 (1992).
Hao, X. L, Sun, J., Wang, A. P., Zettl, A.: Characterization of domains of self-adjoint ordinary diﬀerential operators II. Results in Mathematics, Springer Basel AG, 2011. See also DOI 10.1007/s00025-011-0096-y. M¨ oller, M., Zettl, A.: Symmetric diﬀerential operators and their Friedrichs extension. J. Diﬀerential Equations 115, 50–69 (1995).
Naimark, M. A.: Linear diﬀerential operators II. London Harrap 1968.
Shang, Z. J., Zhu, R. Y.: The domains of self-adjoint extensions of ordinary symmetric diﬀerential operator over ( −∞, ∞). Acta Sci. Natur. Univ. NeiMongGol 17, 17–28 (1986) (In Chinese).
Sun, J.: On the self-adjoint extensions of symmetric ordinary diﬀerential operators with middle deﬁciency indices. Acta Math. Sinica 2, 152–167 (1986).
Sun, J., Wang, W. Y., Zheng, Z. M.: Complex sympletic geometric characterization of self-adjoint domains of singular diﬀerential operators. J. Spectral Math. Appl. 2006.
Wang, A., Sun, J., Zettl, A.: Characterization of domains of self-adjoint ordinary diﬀerential operators. J. Diﬀerential Equations 246, 1600–1622 (2009).
Wang, W. Y., Sun, J.: Complex J-symplectic geometry characterization for J-symmectric extensions of Jsymmectric diﬀerential operators. Beijing: Adv. in Math. 32, 481–484 (2003).
Wei, G. S., Xu, Z. B., Sun, J.: Self-adjoint domains of products of diﬀerential expressions. J. Diﬀ. Equa. 174, 75–90 (2001).
Weidmann, J.: Spectral theory of ordinary diﬀerential operators. Lecture Notes in Math. 1258, Berlin/New York, Springer-Verlag 1987.
Zhang, H. K.: On self-adjointness of the product of two limit-circle diﬀerential operators in vector-function spaces. Acta Scientiarum Naturalium Universitatis NeiMongol 28, 585–591 (1997) (In Chinese).