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

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Page 2149

In view of

measure E * of the preceding lemma may be extended from S ( T ) to a spectral

measure defined on M ( T ) . Then , as in the proof of Theorem 5 , it may be

shown ...

In view of

**condition**( D ) , we have B S M ( T ) . By Theorem 3 . 13 the spectralmeasure E * of the preceding lemma may be extended from S ( T ) to a spectral

measure defined on M ( T ) . Then , as in the proof of Theorem 5 , it may be

shown ...

Page 2153

This will allow us to replace

satisfactory

reflexive space which satisfies ( G ) and whose adjoint satisfies ( B ) is a spectral

operator ...

This will allow us to replace

**condition**( D ) , and in a variety of ways , by moresatisfactory

**conditions**. For example , it will be seen that an operator T in areflexive space which satisfies ( G ) and whose adjoint satisfies ( B ) is a spectral

operator ...

Page 2162

According to Lemma 10

relative to T are dense on to . ... Let T be a bounded linear operator in the B -

space X which satisfies

sets in the ...

According to Lemma 10

**condition**( D ) will be satisfied if the points regularrelative to T are dense on to . ... Let T be a bounded linear operator in the B -

space X which satisfies

**conditions**( B ) and ( G ) and let B be the field of Borelsets in 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