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

### From inside the book

Results 1-3 of 91

Page 1934

PROOF . Using Theorem 2 , we see that if g ( x ) is void , then x ( ) is everywhere

defined , single valued , and hence entire . Since , by VII . 3 . 4 , lim x * x ( ) = lim x

* R ( É ; T ' ) . x = 0 , $ 200 $ 700 it is

PROOF . Using Theorem 2 , we see that if g ( x ) is void , then x ( ) is everywhere

defined , single valued , and hence entire . Since , by VII . 3 . 4 , lim x * x ( ) = lim x

* R ( É ; T ' ) . x = 0 , $ 200 $ 700 it is

**seen**that x * x ( $ ) = 0 for all & and all x * e ...Page 2163

8 , it is

( 6 . ) – F ( E ) B ( 0 , 05 ) SK sup | F ( Xi ) – F ( 81 ) , where the supremum is taken

over those i and j for which o ; o ; is not void . If by the norm ( 7 | is understood ...

8 , it is

**seen**that , for some constant K , 3 F46 . JE ( 0 . ) – É PLE ) E ( 0 ) | FIĚ Ž { P( 6 . ) – F ( E ) B ( 0 , 05 ) SK sup | F ( Xi ) – F ( 81 ) , where the supremum is taken

over those i and j for which o ; o ; is not void . If by the norm ( 7 | is understood ...

Page 2185

Since every function f in C ( A ) is bounded and Borel measurable , the integral sf

( a ) A ( da ) exists and it is

. It will now be shown that A is a spectral measure in X * . By placing f = 1 in ( ii ) ...

Since every function f in C ( A ) is bounded and Borel measurable , the integral sf

( a ) A ( da ) exists and it is

**seen**from ( i ) that ( ii ) S * ( f ) = f ( A ) A ( da ) , feC ( 1 ). It will now be shown that A is a spectral measure in X * . By placing f = 1 in ( ii ) ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

### Other editions - View all

### Common terms and phrases

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