## Linear Operators: Spectral theory |

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Page 918

Suppose that u(enri–8,11) > 0 and let a be an

é,11 such that 0 < u(o) < oo. Let the vectors fi, ..., f"+1 in Št be defined by the

equations f* = [x, , 0, 0,...], f* = [0, X, , 0,...], fn11 - [0, * - - - 0, Zor- 0, - - ..], the

function X ...

Suppose that u(enri–8,11) > 0 and let a be an

**arbitrary**fixed Borel subset of enri–é,11 such that 0 < u(o) < oo. Let the vectors fi, ..., f"+1 in Št be defined by the

equations f* = [x, , 0, 0,...], f* = [0, X, , 0,...], fn11 - [0, * - - - 0, Zor- 0, - - ..], the

function X ...

Page 1179

Theorem 25 remains valid in the full range 1 < p < oo, and for functions f with

values in an

trivial modifications of its proof to functions with values in an

space ...

Theorem 25 remains valid in the full range 1 < p < oo, and for functions f with

values in an

**arbitrary**Hilbert space. PRoof. Note that Theorem 20 goes over withtrivial modifications of its proof to functions with values in an

**arbitrary**Hilbertspace ...

Page 1337

Q.E.D. We have seen in Theorem 1 and Corollary 2 that an

(I) has an expansion of “Fourier integral” type in terms of eigenfunctions W.(t, A)

of the differential operator v. Unfortunately, the interest of Theorem 1 is more ...

Q.E.D. We have seen in Theorem 1 and Corollary 2 that an

**arbitrary**vector f in L2(I) has an expansion of “Fourier integral” type in terms of eigenfunctions W.(t, A)

of the differential operator v. Unfortunately, the interest of Theorem 1 is more ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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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