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

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Page 2125

8 ) / 8 e D ( P ) } and { P ( 8 ) * 18 € D ( P ) } are total on H . It is proved that every g

. s . m . can be extended to a class Do ( P ) of Borel subsets of C which is ...

**Borel sets**and if X , Y EH , then ( P ( 8 ) x , y ) = £ ( P ( x ) x , y ) . ( iv ) The sets { P (8 ) / 8 e D ( P ) } and { P ( 8 ) * 18 € D ( P ) } are total on H . It is proved that every g

. s . m . can be extended to a class Do ( P ) of Borel subsets of C which is ...

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4 that there is a spectral measure A in X * defined on the

that xT ( f ] * = 5 $ ( 1 ) xA ( d ) ) x * ... It will now be shown that , because of the

completeness of B , the operator A ( e ) is , for every

4 that there is a spectral measure A in X * defined on the

**Borel sets**in 1 and suchthat xT ( f ] * = 5 $ ( 1 ) xA ( d ) ) x * ... It will now be shown that , because of the

completeness of B , the operator A ( e ) is , for every

**Borel set**e , the adjoint of a ...Page 2233

... defined as follows . 8 DEFINITION . Let T be a spectral operator with resolution

of the identity E , and let f be a function analytic in an open set U such that E ( U )

= 1 . Let { en } be an arbitrary increasing sequence of bounded

... defined as follows . 8 DEFINITION . Let T be a spectral operator with resolution

of the identity E , and let f be a function analytic in an open set U such that E ( U )

= 1 . Let { en } be an arbitrary increasing sequence of bounded

**Borel sets**with ...### What people are saying - Write a review

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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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### 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