## 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

When it exists , a is in A and ( 31 ) a = S ( s ) e ( ds ) , 121 = ess sup | ã ( s ) ] . ess

Stated otherwise , a = F - 18F where F is the Fourier transform in H and A is the

operation of multiplication by the function à . PROOF . Let any

o ...

When it exists , a is in A and ( 31 ) a = S ( s ) e ( ds ) , 121 = ess sup | ã ( s ) ] . ess

Stated otherwise , a = F - 18F where F is the Fourier transform in H and A is the

operation of multiplication by the function à . PROOF . Let any

**converge**for eacho ...

Page 2218

If a generalized sequence of projections in a o - complete Boolean algebra of

projections in a B - space

strongly . PROOF . In view of Lemma 23 , the proof may be restricted to the case ...

If a generalized sequence of projections in a o - complete Boolean algebra of

projections in a B - space

**converges**weakly to a projection , then it**converges**strongly . PROOF . In view of Lemma 23 , the proof may be restricted to the case ...

Page 2462

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

trace norm , and CT *

XE H | | 2 < 1 } ) is conditionally compact , and thus for each ε > 0 there exists a

finite ...

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

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

**converges**to zero in trace norm . PROOF . The set K = C ( {XE H | | 2 < 1 } ) is conditionally compact , and thus for each ε > 0 there exists a

finite ...

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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