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

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Page 1930

4 we have seen that a bounded normal operator in Hilbert space always has a

uniquely defined bounded and

defined on the field of all Borel subsets of the plane . The operators we shall

study in the ...

4 we have seen that a bounded normal operator in Hilbert space always has a

uniquely defined bounded and

**countably additive**resolution of the identitydefined on the field of all Borel subsets of the plane . The operators we shall

study in the ...

Page 1931

Spectral Theory : Self Adjoint Operators in Hilbert Space Nelson Dunford, Jacob

Theodore Schwartz. → 4 COROLLARY . If the domain of a

spectral measure E is a o - field , then E is

operator ...

Spectral Theory : Self Adjoint Operators in Hilbert Space Nelson Dunford, Jacob

Theodore Schwartz. → 4 COROLLARY . If the domain of a

**countably additive**spectral measure E is a o - field , then E is

**countably additive**in the strongoperator ...

Page 2144

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

is

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

...

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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