## 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

50 ( McCarthy ) Continuing the preceding

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

in the preceding

50 ( McCarthy ) Continuing the preceding

**exercise**, let x4 , xo , and o . be chosenso that ... 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 ...Page 2172

8 ( Fixman ) Let T be as in the preceding

Banach limit as in

83 , . . . ) = ( g ( x ) , 0 , 0 , . . . ) . Show that A2 = 0 and that p ( TX ) = 9 ( x ) and AT

= TA .

8 ( Fixman ) Let T be as in the preceding

**exercise**and let op in ( 1 . ) * be aBanach limit as in

**Exercise**II . 4 . 22 . Let A be defined on lo by Ax = A ( $ 1 , 82 ,83 , . . . ) = ( g ( x ) , 0 , 0 , . . . ) . Show that A2 = 0 and that p ( TX ) = 9 ( x ) and AT

= TA .

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 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 formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm 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