## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 2082

Letting 0 and Y be operators defined on L1 ( 01 , M ) and L . ( 01 , M ) as in the

preceding

the non - separable space L . ( 01 , M ) onto a closed separable subspace sp { E (

0 ) ...

Letting 0 and Y be operators defined on L1 ( 01 , M ) and L . ( 01 , M ) as in the

preceding

**exercise**, show that Yis a one - to - one and bicontinuous mapping ofthe non - separable space L . ( 01 , M ) onto a closed separable subspace sp { E (

0 ) ...

Page 2172

Show that T is a spectral operator of class ( I ' ) , but that it is not a spectral

operator of class ( X * ) . 8 ( Fixman ) Let T be as in the preceding

op in ( 1 . ) * be a Banach limit as in

Ax ...

Show that T is a spectral operator of class ( I ' ) , but that it is not a spectral

operator of class ( X * ) . 8 ( Fixman ) Let T be as in the preceding

**exercise**and letop in ( 1 . ) * be a Banach limit as in

**Exercise**II . 4 . 22 . Let A be defined on lo byAx ...

Page 2489

1 / 2 ( Hint : Use ( c ) and ( e ) of the preceding

adjoint operator in Hilbert space . Let V be an operator of trace class , and put H ,

= H + V . Put U : = exp ( itH , ) exp ( - itH ) . Let we { ac ( H ) and | | W | | # < 00 ( cf .

1 / 2 ( Hint : Use ( c ) and ( e ) of the preceding

**exercise**. ) 19 Let H be a selfadjoint operator in Hilbert space . Let V be an operator of trace class , and put H ,

= H + V . Put U : = exp ( itH , ) exp ( - itH ) . Let we { ac ( H ) and | | W | | # < 00 ( cf .

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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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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 countably additive defined Definition denote dense determined differential operator 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