## 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). Since 0 < —f(t) ...

—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). Since 0 < —f(t) ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero