Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-08T12:09:23.127Z Has data issue: false hasContentIssue false
  • English
  • Français

20.—Deficiency Indices of Polynomials in Symmetric Differential Expressions, II*†

Published online by Cambridge University Press:  14 February 2012

Anton Zettl
Anton Zettl
Affiliation:
Northern Illinois University, De Kalb, Illinois, and University of Dundee.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Given a symmetric (formally self-adjoint) ordinary linear differential expression L which is regular on the interval [0, ∞) and has C∞ coefficients, we investigate the relationship between the deficiency indices of L and those of p(L), where p(x) is any real polynomial of degree k > 1. Previously we established the following inequalities: (a) For k even, say k = 2r, N+(p(L)), N−(p(L)) ≧ r[N+(L)+N−(L)] and (b) for k odd, say k = 2r+1

where N+(M), N−(M) denote the deficiency indices of the symmetric expression M (or of the minimal operator associated with M in the Hilbert space L2(0, ∞)) corresponding to the upper and lower half-planes, respectively. Here we give a necessary and sufficient condition for equality to hold in the above inequalities.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 73 , 1975 , pp. 301 - 306
Copyright
Copyright © Royal Society of Edinburgh 1975

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Akhiezer, N. I. and Glazman, I. M., Theory of Linear Operators in Hilbert Space, II. New York: Ungar, 1963.Google Scholar
2Chaudhuri, J. and Everitt, W. N., On the square of a formally self-adjoint differential expression. J.London Math. Soc, 1, 661–673, 1969.CrossRefGoogle Scholar
3Dunford, N. and Schwartz, J. T., Linear Operators, II. New York: Interscience, 1963.Google Scholar
4Everitt, W. N. and Giertz, M., On some properties of the powers of a formally self-adjoint differential expression. Proc. London Math. Soc, 24, 149–170, 1972.CrossRefGoogle Scholar
5Everitt, W. N., Integrable-square solutions of ordinary differential equations. Quart. J. Math., 10, 145–155, 1959.CrossRefGoogle Scholar
6Glazman, I. M., On the theory of singular differential operators. Uspehi Mat. Nauk, 5 (6/40), 102–135, 1950. (Amer. Math. Soc. Transl., 96, 1953.)Google Scholar
7Goldberg, S., Unbounded Linear Operators. New York: McGraw-Hill, 1966.Google Scholar
8Kauffman, R. M., Polynomials and the limit point condition. Trans. Amer. Math. Soc, 201, 347–366, 1975.CrossRefGoogle Scholar
9Miller, K. S., Linear Differential Equations in the Real Domain. New York: Norton, 1963.Google Scholar
10Naimark, M. A., Linear Differential Operators, II. New York: Ungar, 1968.Google Scholar
11Zettl, , Anton, , The limit-point and limit-circle cases for polynomials in a differential operator, Proc. Royal Soc. Edinburgh, Sect. A, 72, 219–224, 1975.CrossRefGoogle Scholar
12Zettl, , Anton, , Deficiency indices of polynomials in symmetric differential expressions. Lecture Notes in Mathematics, 415.Google Scholar