## Linear Operators: Spectral theory |

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Page 1215

... ( ds ) exists in the mean square sense and equals ( UD ) . ( 2 ) , proving ( c ) . Q

. E . D . Using the notation of the

Lemma 9 , in a ( 2 ) W . ( : , 2 ) ua ( da ) = E ( [ - n , n ] ) F ( T ) a → F ( T ) a = U ; F =

fa .

... ( ds ) exists in the mean square sense and equals ( UD ) . ( 2 ) , proving ( c ) . Q

. E . D . Using the notation of the

**preceding**proof we let F = ( Uf ) , so that , byLemma 9 , in a ( 2 ) W . ( : , 2 ) ua ( da ) = E ( [ - n , n ] ) F ( T ) a → F ( T ) a = U ; F =

fa .

Page 1480

Under the hypotheses and with the notation of the

defined by at most one boundary condition , which is a boundary condition at an

end point a of the interval I . Let < 2o , and let ost , a ) be a solution of the

equation to ...

Under the hypotheses and with the notation of the

**preceding**theorem , T isdefined by at most one boundary condition , which is a boundary condition at an

end point a of the interval I . Let < 2o , and let ost , a ) be a solution of the

equation to ...

Page 1771

... XEI , such that uột , 2 ) = 0 , ( 24 ( 1 , 2 ) = . . . = 0 - 1 ( 2 ) u ( 1 , 2 ) = 0 , t > 0 , 62

, and such that lim ( u ( t , : ) - 1 ( : ) ] = 0 . t - 0 Proof . Statement ( i ) follows from

the

statement ...

... XEI , such that uột , 2 ) = 0 , ( 24 ( 1 , 2 ) = . . . = 0 - 1 ( 2 ) u ( 1 , 2 ) = 0 , t > 0 , 62

, and such that lim ( u ( t , : ) - 1 ( : ) ] = 0 . t - 0 Proof . Statement ( i ) follows from

the

**preceding**theorem and Theorem 6 . 23 . Statement ( ii ) follows fromstatement ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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