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Proceedings of the American Mathematical Society

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

 
 

 

Separation for differential operators and the $ L\sp{p}$ spaces


Author: Anton Zettl
Journal: Proc. Amer. Math. Soc. 55 (1976), 44-46
DOI: https://doi.org/10.1090/S0002-9939-1976-0393646-7
MathSciNet review: 0393646
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Abstract: The differential operator

$\displaystyle My = {y^{(n)}} + {q_{n - 1}}{y^{(n - 1)}} + \cdots + {q_1}y' + {q_0}y$

is said to be separated in $ {L^p}(0,\infty )$ if $ y \in {L^p}(0,\infty ),{y^{(n - 1)}}$ absolutely continuous, and $ My \in {L^p}(0,\infty )$ imply that $ {q_i}{y^{(i)}} \in {L^p}(0,\infty )$ for all $ i = 0,1, \ldots ,n - 1$. As a special case of our main result we obtain: $ M$ is separated in $ {L^p}(0,\infty )$ if $ {q_i} = {b_i} + {s_i}$ where $ {b_i}$ is essentially bounded and $ {s_i} \in {L^{pi}}(0,\infty )$ for some $ {p_i} \geqslant p,i = 0,1, \ldots ,n - 1$. The case $ n = 2,p = 2,{q_1} \equiv 0$ is due to Everitt-Giertz-Weidmann [2]. In the same paper these authors show that this result is best possible in the sense that for any $ p$ satisfying $ 1 < p < 2$ a function $ q$ exists in $ {L^p}(0,\infty )$ such that $ y'' + qy$ is not separated in $ {L^2}(0,\infty )$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0393646-7
Keywords: Ordinary linear differential operators, separation of differential operators, $ {L^p}$ spaces
Article copyright: © Copyright 1976 American Mathematical Society

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