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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

General theory of the factorization of ordinary linear differential operators


Author: Anton Zettl
Journal: Trans. Amer. Math. Soc. 197 (1974), 341-353
MSC: Primary 34A30
DOI: https://doi.org/10.1090/S0002-9947-1974-0364724-6
MathSciNet review: 0364724
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Abstract: The problem of factoring the general ordinary linear differential operator $ Ly = {y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y$ into products of lower order factors is studied. The factors are characterized completely in terms of solutions of the equation $ Ly = 0$ and its adjoint equation $ {L^ \ast }y = 0$. The special case when L is formally selfadjoint of order $ n = 2k$ and the factors are of order k and adjoint to each other reduces to a well-known result of Rellich and Heinz: $ L = {Q^ \ast }Q$ if and only if there exist solutions $ {y_1}, \cdots ,{y_k}$ of $ Ly = 0$ satisfying $ W({y_1}, \cdots ,{y_k}) \ne 0$ and $ [{y_i};{y_j}] = 0$ for $ i,j = 1, \cdots ,k$; where [ ; ] is the Lagrange bilinear form of L.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0364724-6
Keywords: Linear ordinary differential equations, factoring differential operators, property W, conjugate solutions, Wronskians, disconjugacy
Article copyright: © Copyright 1974 American Mathematical Society