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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.
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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

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References

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