## Linear Operators: Spectral theory |

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Page 1085

0 (–1)" A m! det : R An-1 n-1 48 (Fredholm Determinant Series) Let the

hypotheses of the

space of that exercise is L2(S, 2, u), where (S, 2, u) is a positive measure space.

Let A(* ...

0 (–1)" A m! det : R An-1 n-1 48 (Fredholm Determinant Series) Let the

hypotheses of the

**preceding**exercise be satisfied, and suppose that the Hilbertspace of that exercise is L2(S, 2, u), where (S, 2, u) is a positive measure space.

Let A(* ...

Page 1215

... by continuity, J. f(s)\WSorūs) = (U.), - (Us). Thus ssf(s) W.(s, 2),(ds) exists in the

mean square sense and equals (Uf),(A), proving (c). Q.E.D. Using the notation of

the

... by continuity, J. f(s)\WSorūs) = (U.), - (Us). Thus ssf(s) W.(s, 2),(ds) exists in the

mean square sense and equals (Uf),(A), proving (c). Q.E.D. Using the notation of

the

**preceding**proof we let F = (Uf), so that, by Lemma 9, s". (UI),(2)W.G., ou,(da) ...Page 1419

—f(t). and. f(t). = +f(2s,. 1–t). We. have. (-f)". = q(–f),. f. = qif,. where qi(t) = q(2s, 1.1

—t) > q(t), since q is monotone decreasing. By the

s, 11, miri). In particular —f(miri)= f(miri) “ fi(mori) ...

—f(t). and. f(t). = +f(2s,. 1–t). We. have. (-f)". = q(–f),. f. = qif,. where qi(t) = q(2s, 1.1

—t) > q(t), since q is monotone decreasing. By the

**preceding**lemma, -f(t) < f(t) in [s, 11, miri). In particular —f(miri)= f(miri) “ fi(mori) ...

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

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