## 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 1756

Hence we find that if \ yl < r , fly ) = 0 , and

uniqueness of the function V of the theorem is an evident consequence of

existence ...

Hence we find that if \ yl < r , fly ) = 0 , and

**statement**( i ) is fully proved . ( B ) Theuniqueness of the function V of the theorem is an evident consequence of

**statement**( i ) . Moreover ,**statement**( i ) enables us to reduce the proof of theexistence ...

Page 1771

... such that lim | u ( t , : ) - | ( : ) = 0 . t > 0 , 62 , to PROOF .

from the preceding theorem and Theorem 6 . 23 .

hypotheses of ...

... such that lim | u ( t , : ) - | ( : ) = 0 . t > 0 , 62 , to PROOF .

**Statement**( i ) followsfrom the preceding theorem and Theorem 6 . 23 .

**Statement**( ii ) follows from**statement**( ii ) of the preceding theorem , since a function satisfying thehypotheses of ...

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 other sections not shown

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additive adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently consider constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function 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 satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero