## 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 Elis ( 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 a

...

It follows from equations ( iv ) and ( v ) of Lemma 3 that Elis ( 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 a

...

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

self adjoint operator in Hilbert space . ... If a is not real , an expansion of the

scalar product ( ( aI - T ' ) x , ( al - T ) x )

R ...

These lemmas will

**show**that the hypotheses of Theorem 5.18 are satisfied by aself adjoint operator in Hilbert space . ... If a is not real , an expansion of the

scalar product ( ( aI - T ' ) x , ( al - T ) x )

**shows**that llal – T ' ) x | 2 = \ I ( a ) x12 + | (R ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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### Common terms and phrases

adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator discrete domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero