General theory of the factorization of ordinary linear differential operators

Zettl, Anton

doi:10.1090/S0002-9947-1974-0364724-6

General theory of the factorization of ordinary linear differential operators
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by Anton Zettl

Trans. Amer. Math. Soc. 197 (1974), 341-353

DOI: https://doi.org/10.1090/S0002-9947-1974-0364724-6

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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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