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

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Page 2010

3 that pl ( Â ( s ) ) = 0 almost everywhere on S . Thus , for almost all s , o

on the spectrum o ( Â ( s ) ) . So , for some set o , in { with eloo ) = e , the function

o

3 that pl ( Â ( s ) ) = 0 almost everywhere on S . Thus , for almost all s , o

**vanishes**on the spectrum o ( Â ( s ) ) . So , for some set o , in { with eloo ) = e , the function

o

**vanishes**on Uses , O ( Â ( s ) ) , and since y is continuous , it also**vanishes**on ...Page 2046

We shall now show that it is a semi - group , that is , 8 ( $ 1 + $ 2 ) = S ( $ 1 ) S ( $

2 ) if R ( $ 1 ) > 0 and R ( % 2 ) > 0 . For any q in Hp and t > 0 the function ( 81 ) = [

S8 + t ) – S ( 1 ) S ( ) ] is analytic and

We shall now show that it is a semi - group , that is , 8 ( $ 1 + $ 2 ) = S ( $ 1 ) S ( $

2 ) if R ( $ 1 ) > 0 and R ( % 2 ) > 0 . For any q in Hp and t > 0 the function ( 81 ) = [

S8 + t ) – S ( 1 ) S ( ) ] is analytic and

**vanishes**for $ 1 on the positive real axis .Page 2161

13 ) , suffice to show that the functional x * = 0 ) is the only functional which

* * = 0 . To see that x * = 0 it will first be shown that ( ii ) ( iii ) * * € ( 101 * – T * ) NX

* .

13 ) , suffice to show that the functional x * = 0 ) is the only functional which

**vanishes**on the manifold ( i ) . Let 2 * be such a functional . Then ( 101 * – T ' * ) ** * = 0 . To see that x * = 0 it will first be shown that ( ii ) ( iii ) * * € ( 101 * – T * ) NX

* .

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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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analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero