zbMATH — the first resource for mathematics

Self-adjoint differential vector-operators and matrix Hilbert spaces. I. (English) Zbl 1236.47042
Summary: In the current work, a generalization of the famous Weyl-Kodaira inversion formulas for the case of selfadjoint differential vector operators is proved. A formula for spectral resolutions over an analytical defining set of solutions is discussed. The article is the first part of the planned two-part survey on the structural spectral theory of selfadjoint differential vector operators in matrix Hilbert spaces.
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
34L05 General spectral theory of ordinary differential operators
47A10 Spectrum, resolvent
47B25 Linear symmetric and selfadjoint operators (unbounded)
Full Text: DOI
[1] M.S. Sokolov: “An abstract approach to some spectral problems of direct sum differential operators”, Electronic. J. Diff. Eq., Vol. 75, (2003), pp. 1-6. · Zbl 1045.47016
[2] M.S. Sokolov: “On some spectral properties of operators generated by multi-interval quasi-differential systems”, Methods Appl. Anal., Vol. 10(4), (2004), pp. 513-532. · Zbl 1078.34066
[3] R.R. Ashurov and M.S. Sokolov: “On spectral resolutions connected with self-adjoint differential vector-operators in a Hilbert space”, Appl. Anal., Vol. 84(6), (2005), pp. 601-616. http://dx.doi.org/10.1080/00036810500048160 · Zbl 1086.34067
[4] W.N. Everitt and A. Zettl: “Quasi-differential operators generated by a countable number of expressions on the real line”, Proc. London Math. Soc., Vol. 64(3), (1992), pp. 524-544. · Zbl 0723.34022
[5] W.N. Everitt and L. Markus: “Multi-interval linear ordinary boundary value problems and complex symplectic algebra”, Mem. Am. Math. Soc., Vol. 715, (2001). · Zbl 0982.47032
[6] R.R. Ashurov and W.N. Everitt: “Linear operators generated by a countable number of quasi-differential expressions”, Appl. Anal., Vol. 81(6), (2002), pp. 1405-1425. http://dx.doi.org/10.1080/0003681021000035506 · Zbl 1049.47043
[7] M.A. Naimark: Linear differential operators, Ungar, New York, 1968.
[8] M. Reed and B. Simon: Methods of modern mathematical physics, Vol. 1: Functional Analysis, Academic Press, New York, 1972. · Zbl 0242.46001
[9] N. Dunford and J.T. Schwartz: Linear operators, Vol. 2: Spectral Theory, Interscience, New York, 1964. · Zbl 0243.47001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.