## Linear Operators: Spectral theory |

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Page 1021

preceding remarks that the final conclusion of the lemma is equivalent to the

since W is unitary , det ( A ) = det ( W - 1 AW ) and ( A - 1x , y ) = ( A - 1 Won , y ) =

( W - 1 ...

preceding remarks that the final conclusion of the lemma is equivalent to the

**statement**that det ( A ) ( A - 1x , y ) = \ y | | | A | | - 1 ( n - 1 ) - ( n - 1 ) / 2 . Now ,since W is unitary , det ( A ) = det ( W - 1 AW ) and ( A - 1x , y ) = ( A - 1 Won , y ) =

( W - 1 ...

Page 1463

b ) is an evident consequence of

illustrated by the pair of functions sin it and sin ut for u > 2 ; the corollary by the

pair of ...

**Statement**( a ) is the special case T1 = ty of the preceding theorem .**Statement**(b ) is an evident consequence of

**statement**( a ) . Q . E . D . The theorem isillustrated by the pair of functions sin it and sin ut for u > 2 ; the corollary by the

pair of ...

Page 1771

function satisfying the hypotheses of the present

Theorem 6 .

**Statement**( i ) follows from the preceding theorem and Theorem 6 . 23 .**Statement**( ii ) follows from**statement**( ii ) of the preceding theorem , since afunction satisfying the hypotheses of the present

**statement**( ii ) evidently ( cf .Theorem 6 .

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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additive adjoint operator Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex 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 matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero