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

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Page 2144

It clearly preserves finite disjoint unions , takes complements into complements ,

is countably additive in the X

show that A ( 0 ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for

...

It clearly preserves finite disjoint unions , takes complements into complements ,

is countably additive in the X

**topology**of X * , and is bounded . It remains only toshow that A ( 0 ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for

...

Page 2203

The mapping E H0 ( E ) is clearly an isomorphism between B and the Boolean

algebra of all open and closed subsets of 1 . These open and closed sets o ( E )

form a basis for the

The mapping E H0 ( E ) is clearly an isomorphism between B and the Boolean

algebra of all open and closed subsets of 1 . These open and closed sets o ( E )

form a basis for the

**topology**in 1 . To see this , note that sets of the form { a \ | ( T ...Page 2281

The results are parallel to those in X . The weaker form of completeness enjoyed

by B * leads to a weaker form of the representation theorem , as the relevant

convenience ...

The results are parallel to those in X . The weaker form of completeness enjoyed

by B * leads to a weaker form of the representation theorem , as the relevant

**topology**in X * is now the X -**topology**. As before we shall suppose forconvenience ...

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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