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

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Page 1930

3 DEFINITION . A spectral measure E is said to be countably additive if for each 2

* in X * and each x in X the scalar set function x * E ( • ) x is countably additive on

the

3 DEFINITION . A spectral measure E is said to be countably additive if for each 2

* in X * and each x in X the scalar set function x * E ( • ) x is countably additive on

the

**domain**of E . → 4 COROLLARY . If the**domain**of a countably 1930 XV . 2 .Page 1931

4 COROLLARY . If the

field , then E is countably additive in the strong operator topology and bounded .

The boundedness of E ( 0 ) follows from Corollaries IV . 10 . 2 and II . 3 . 21 .

4 COROLLARY . If the

**domain**of a countably additive spectral measure E is a o -field , then E is countably additive in the strong operator topology and bounded .

The boundedness of E ( 0 ) follows from Corollaries IV . 10 . 2 and II . 3 . 21 .

Page 2256

where C , is a finite collection of closed Jordan curves bounding a

containing the union of o ( T ' ) and a neighborhood of infinity , C , being oriented

in the customary positive sense of complex variable theory . The curves C of the ...

where C , is a finite collection of closed Jordan curves bounding a

**domain**Dicontaining the union of o ( T ' ) and a neighborhood of infinity , C , being oriented

in the customary positive sense of complex variable theory . The curves C of the ...

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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