## 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 1994

Let any

Theorem II . 3 . 6 proves ( 29 ) . ... F - 11n X , and since Ano

from the continuity of F that - Am ( 8 ) | 2 ds = 0 , m . no which implies ( III . 3 . 6 )

that in ...

Let any

**converge**for each o in H . Theorem 4 shows that lanl = nloo , andTheorem II . 3 . 6 proves ( 29 ) . ... F - 11n X , and since Ano

**converges**, we seefrom the continuity of F that - Am ( 8 ) | 2 ds = 0 , m . no which implies ( III . 3 . 6 )

that in ...

Page 2218

It must be shown that { Ex }

so a consideration of the sequence { Ea – E } shows that it may be assumed that

E = 0 . Thus , to make an indirect proof , it is assumed that the sequence { Ex } is ...

It must be shown that { Ex }

**converges**strongly to E . By Lemma 6 , E is in B andso a consideration of the sequence { Ea – E } shows that it may be assumed that

E = 0 . Thus , to make an indirect proof , it is assumed that the sequence { Ex } is ...

Page 2462

Moreover , if C belongs to the trace class C1 , then T , C

trace norm , and CT *

xe H | 1 } ) is conditionally compact , and thus for each € > 0 there exists a finite

set ...

Moreover , if C belongs to the trace class C1 , then T , C

**converges**to zero intrace norm , and CT *

**converges**to zero in trace norm . PROOF . The set K = C ( {xe H | 1 } ) is conditionally compact , and thus for each € > 0 there exists a finite

set ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

28 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 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