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Equivalence, adjoints and symmetry of quasi-differential expressions with matrix-valued coefficients and polynomials in them

Published online by Cambridge University Press:  14 November 2011

Hilbert Frentzen
Affiliation:
Fachbereich 6, Mathematik, Universitat Essen-GHS, D-4300 Essen 1, B.R.D.

Synopsis

Quasi-differential expressions with matrix-valued coefficients, which generalize those of Shin and Zettl, are considered with regard to equivalence, adjoints and symmetry. The characterization results imply that in the scalar case the class of quasi-differential expressions considered here coincides with that of Shin and is equivalent to that of Zettl. Furthermore polynomials in quasi-differential expressions are defined as expressions of the same kind and shown to coincide with the usual ones. Finally it is indicated that the known general results for the deficiency indices carry over to quasi-differential expressions.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1982

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References

1Achieser, N. I. and Glazman, I. M.. Theory of linear operators in Hilbert space (New York: Ungar, 1963).Google Scholar
2Atkinson, F. V.. Discrete and continuous boundary value problems (New York: Academic Press, 1964).Google Scholar
3Everitt, W. N. and Zettl, A.. Products of differential expressions without smoothness assumptions. Quaestiones Math. 3 (1978), 6782.CrossRefGoogle Scholar
4Everitt, W. N. and Zettl, A.. Generalized symmetric ordinary differential expressions I: the general theory. Nieuw Arch. Wisk. 27 (1979), 363397.Google Scholar
5Frentzen, H.. On the left-definiteness of S-hermitian systems of differential equations. Quaestiones Math. 3 (1979), 281312.CrossRefGoogle Scholar
6Glazman, I. M.. On the theory of differential operators. Uspehi Mat. Nauk 40 (1950), 102135; English translation Amer. Math. Soc. Transl. (1) 4 (1962), 331–372.Google Scholar
7Halperin, I.. Closures and adjoints of linear differential operators. Ann. of Math. 38 (1937), 880919.CrossRefGoogle Scholar
8Hamburger, H. L.. Remarks on self-adjoint differential operators. Proc. London Math. Soc. 3 (1953), 446463.CrossRefGoogle Scholar
9Kauffman, R. M., Read, T. T. and Zettl, A.. The deficiency index problem for powers of ordinary differential expressions. Lecture Notes in Mathematics 621 (Berlin: Springer, 1977).Google Scholar
10Kogan, V. I. and Rofe-Beketov, F. S.. On the square-integrable solutions of symmetric systems of differential equations of arbitrary order. Proc. Roy. Soc. Edinburgh Sect. A 74 (1976), 540.CrossRefGoogle Scholar
11Naimark, M. A.. Linear differential operators, part II (New York: Ungar, 1968).Google Scholar
12Reid, W. T.. Adjoint linear differential operators. Trans. Amer. Math. Soc. 85 (1957), 446461.CrossRefGoogle Scholar
13Reid, W. T.. A class of two-point boundary problems. Illinois J. Math. 2 (1958), 434453.CrossRefGoogle Scholar
14Reid, W. T.. Ordinary differential equations (New York: Wiley, 1971).Google Scholar
15Shin, D.. Existence theorems for the quasi-differential equation of the nth order. Dokl. Akad. Nauk SSSR 18 (1938), 515518.Google Scholar
16Shin, D.. On the solutions of the self-adjoint differential equation u [n[ = I(l) ≠ 0 in L 2(0,∞). Dokl. Akad. Nauk SSSR 18 (1938), 519522.Google Scholar
17Shin, D.. On the quasi-differential transformations in Hilbert space. Dokl. Akad Nauk SSSR 18 (1938), 523526.Google Scholar
18Walker, P. W.. A vector-matrix formulation for formally symmetric ordinary differential equations with applications to solutions of integrable square. J. London Math. Soc. 9 (1974), 151159.Google Scholar
19Zettl, A.. Formally self-adjoint quasi-differential operators. Rocky Mountain J. Math. 5 (1975), 453474.Google Scholar
20Zettl, A. Powers of real symmetric differential expressions without smoothness assumptions. Quaestiones Math. 1 (1976), 8394.CrossRefGoogle Scholar