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On the commutativity of certain quasi-differential expressions II

Published online by Cambridge University Press:  14 November 2011

H. Frentzen ,
D. Race  and
A. Zettl
H. Frentzen
Affiliation:
Fachbereich 6-Mathematik und Informatik, Universität Essen, Universitätsstr.3, W-4300 Essen 1, Germany
D. Race
Affiliation:
Mathematics Department, University of Surrey, Guildford, Surrey GU2 5XH, U.K.
A. Zettl
Affiliation:
Department of Mathematics, Northern Illinois University, DeKalb, Illinois 60115, U.S.A,
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Synopsis

We consider the question: when do two ordinary, linear, quasi-differential expressions commute? For classical differential expressions, answers to this question are well known. The set of all expressions which commute with a given such expression form a commutative ring. For quasi-differential expressions less is known and such an algebraiastructure can no longer be exploited. Using the theory of very general quasi-differential expressions with matrix-valued coefficients, we prove some general results concerning commutativity of such expressions. We show how, when specialised to scalar expressions, these results include a proof of the conjecture that if a 2nth-order scalar, J-symmetric (or real symmetric) quasi-differential expression commutes with a second order expression having the same properties, then the former must be an nth-order polynomial in the latter. This result was conjectured in a paper by Race and Zettl, to which this paper is a sequel.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 1 , 1993 , pp. 27 - 43
Copyright
Copyright © Royal Society of Edinburgh 1993

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