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Transactions of the American Mathematical Society

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



Some new classes of kernels whose Fredholm determinants have order less than one

Author: Dale W. Swann
Journal: Trans. Amer. Math. Soc. 160 (1971), 427-435
MSC: Primary 45.11
MathSciNet review: 0283513
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Abstract: Let $ K(s,t)$ be a complex-valued $ {L_2}$ kernel on the square $ a \leqq s,t \leqq b$ and $ \{ {\lambda _v}\} $, perhaps empty, denote the set of finite characteristic values (f.c.v.) of $ K$, arranged according to increasing modulus. Such f.c.v. are complex numbers appearing in the integral equation $ {\phi _v}(s) = {\lambda _v}\int_a^b {K(s,t){\phi _v}(t)dt} $, where the $ {\phi _v}(s)$ are nontrivial $ {L_2}$ functions on $ [a,b]$. Further let $ {k_1} = \int_a^b {K(s,s)} $ be well defined so that the Fredholm determinant of $ K,D(\lambda )$, exists, and let $ \mu $ be the order of this entire function. It is shown that (1) if $ K(s,t)$ is a function of bounded variation in the sense of Hardy-Krause, then $ \mu \leqq 1$; (2) if in addition to the assumption (1), $ K(s,t)$ satisfies a uniform Lipschitz condition of order $ \alpha > 0$ with respect to either variable, then $ \mu < 1$ and $ {k_1} = {\Sigma _v}1/{\lambda _v}$; (3) if $ K(s,t)$ is absolutely continuous as a function of two variables and $ {\partial ^2}K/\partial s\partial t$ (which exists almost everywhere) belongs to class $ {L_p}$ for some $ p > 1$, then $ \mu < 1$ and $ {k_1} = {\Sigma _v}1/{\lambda _v}$. In (2) and (3), the condition $ {k_1} \ne 0$ implies $ K(s,t)$ possesses at least one f.c.v.

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Keywords: $ {L_2}$ kernels, characteristic values, integral equations, Fredholm determinant, Lipschitz (Hölder) conditions, functions of bounded variation in Hardy-Krause sense, absolutely continuous functions of two variables
Article copyright: © Copyright 1971 American Mathematical Society

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