## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2313

Any element Ef can be written uniquely as Ef = 3 0 , 01 ( S ) , where the oilf ) are

linear functionals . ... To see that 4 * , 4 * , . . . , 4 are

1 B 4 * = 0 ; then P , = ( 847 ) ys = 0 , so that Lemma 9 is completely proved .

Any element Ef can be written uniquely as Ef = 3 0 , 01 ( S ) , where the oilf ) are

linear functionals . ... To see that 4 * , 4 * , . . . , 4 are

**linearly independent**, let r =1 B 4 * = 0 ; then P , = ( 847 ) ys = 0 , so that Lemma 9 is completely proved .

Page 2327

it follows that there exist at least two

equation to = do q such that Bi ( 91 ) = B : ( 92 ) = 0 , i = 1 , . . . , k . Then 21 may

be represented uniquely as 91 = } = 1 C1 ; 0 ; ( u ( a ) ) and , similarly , P2 = ) = 1 ...

it follows that there exist at least two

**linearly independent**solutions P1 , 92 of theequation to = do q such that Bi ( 91 ) = B : ( 92 ) = 0 , i = 1 , . . . , k . Then 21 may

be represented uniquely as 91 = } = 1 C1 ; 0 ; ( u ( a ) ) and , similarly , P2 = ) = 1 ...

Page 2394

On the other hand , since these two solutions are

have a linear relation o2 = aới + boy . It is clear from Lemma 1 that such a linear

combination can only have the indicated asymptotic form if a = 1 , b = c ( u ( a ) ) .

On the other hand , since these two solutions are

**linearly independent**, we musthave a linear relation o2 = aới + boy . It is clear from Lemma 1 that such a linear

combination can only have the indicated asymptotic form if a = 1 , b = c ( u ( a ) ) .

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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