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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On factorizations of selfadjoint ordinary differential operators


Author: Antonio Granata
Journal: Proc. Amer. Math. Soc. 86 (1982), 260-266
MSC: Primary 47E05; Secondary 34B25
MathSciNet review: 667285
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Abstract: Consider an ordinary linear differential operator $ L$, of order $ n \geqslant 1$, represented by $ Lu \equiv {a_n}(t){u^{(n)}} + \cdots + {a_0}(t)u\;\forall u \in {C^n}(a,b)$, with real-valued coefficients $ {a_k} \in {C^k}(a,b)$, $ 0 \leqslant k \leqslant n$, $ {a_n} \ne 0$ on $ (a,b)$. According to a classical result, if $ L$ is formally selfadjoint on $ (a,b)$ then it has a factorization of the type $ Lu \equiv {p_n}[{p_{n - 1}}( \cdots ({p_1}({p_0}u)')' \cdots )']'\forall u \in {C^n}(a,b)$, where the $ {p_k}$'s are sufficiently-smooth and everywhere nonzero functions on $ (a,b)$ such that $ {p_k} = {p_{n - k}}$ $ (k = 0, \ldots ,n)$. In this note we shall examine this result critically and show by means of counterexamples that the different classical proofs are either merely local or purely heuristic. A proof, which is both rigorous and global, is inferred from recent results on canonical factorizations of disconjugate operators. In addition, information is obtained on the behavior of the $ {p_k}$'s at the endpoints of $ (a,b)$ which may prove useful in applications.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0667285-7
Keywords: Selfadjoint, canonical factorizations
Article copyright: © Copyright 1982 American Mathematical Society