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

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Results 1-3 of 90

Page 1979

It follows from equations ( iv ) and ( v ) of Lemma 3 that Ey ( s ) ; Â ( s ) ) is e -

essentially bounded on S . Lemma 4 then

is satisfied . Q . E . D . 8 COROLLARY . Every operator A in AP is the strong limit

of ...

It follows from equations ( iv ) and ( v ) of Lemma 3 that Ey ( s ) ; Â ( s ) ) is e -

essentially bounded on S . Lemma 4 then

**shows**that condition ( i ) of the theoremis satisfied . Q . E . D . 8 COROLLARY . Every operator A in AP is the strong limit

of ...

Page 2169

This

continuous function g . A repetition of this argument

and g are both bounded Borel functions . Thus the operators f ( T ) and g ( T )

commute and ...

This

**shows**that ( vi ) holds for every bounded Borel function f and everycontinuous function g . A repetition of this argument

**shows**that it also holds if fand g are both bounded Borel functions . Thus the operators f ( T ) and g ( T )

commute and ...

Page 2170

These lemmas will

... If a is not real , an expansion of the scalar product ( ( al – T ) » , ( al – T ' ) x )

...

These lemmas will

**show**that the hypotheses of Theorem 5 . 18 are satisfied by a... If a is not real , an expansion of the scalar product ( ( al – T ) » , ( al – T ' ) x )

**shows**that | ( al — T ' ) < / 2 = \ I ( ) a [ 2 + | ( R ( Q ) I – T ) « [ ? 2 | ( a ) / 2 [ 2 / 2 , so...

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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