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Transactions of the American Mathematical Society

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

 

 

Ordinary differential operators under Stieltjes boundary conditions


Authors: Richard C. Brown and Allan M. Krall
Journal: Trans. Amer. Math. Soc. 198 (1974), 73-92
MSC: Primary 47E05; Secondary 34B10
DOI: https://doi.org/10.1090/S0002-9947-1974-0358436-2
MathSciNet review: 0358436
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Abstract: The operator $ {L_p}y = y' + Py$, whose domain is determined in part by the Stieltjes integral boundary condition $ \int_0^1 {d\nu (t)y(t) = 0} $, is studied in $ \mathcal{L}_n^p(0,1),1 \leqslant p < \infty $. It is shown that $ {L_p}$ has a dense domain; hence there exists a dual operator $ L_q^ + $ operating on $ \mathcal{L}_n^q(0,1)$. After finding $ L_q^ + $ we show that both $ {L_p}$ and $ L_q^ + $ are Fredholm operators. This implies some elementary results concerning the spectrum and states of $ {L_p}$. Finally two eigenfunction expansions are derived.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0358436-2
Article copyright: © Copyright 1974 American Mathematical Society