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

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Page 2137

Even though ( A ) is taken as a standing assumption throughout this section , it

will be indicated parenthetically in the statement of each lemma in the

which it is used . 1 LEMMA ( A ) . If a , ß are complex numbers and x , y are

vectors in ...

Even though ( A ) is taken as a standing assumption throughout this section , it

will be indicated parenthetically in the statement of each lemma in the

**proof**ofwhich it is used . 1 LEMMA ( A ) . If a , ß are complex numbers and x , y are

vectors in ...

Page 2218

algebra A generated by the projections in its resolution of the identity , it suffices

to show that every operator in A is a spectral operator of scalar type . This follows

...

**Proof**. Since a spectral operator of scalar type is clearly in the weakly closedalgebra A generated by the projections in its resolution of the identity , it suffices

to show that every operator in A is a spectral operator of scalar type . This follows

...

Page 2281

The

representation of E * X * when E * € & * and has finite uniform multiplicity n . The

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

...

The

**proof**is similar to that of Lemma 17 and will be omitted . We turn now to therepresentation of E * X * when E * € & * and has finite uniform multiplicity n . The

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

...

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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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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