## 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 2256

Then T is a spectral operator if and only if ( a ) the family of projections E ( o ; T )

corresponding to compact spectral sets of T is uniformly bounded , and ( b ) no

non -

o ...

Then T is a spectral operator if and only if ( a ) the family of projections E ( o ; T )

corresponding to compact spectral sets of T is uniformly bounded , and ( b ) no

non -

**zero**x in X satisfies the equation E ( o ) x = 0 for every compact spectral seto ...

Page 2325

0 3 Rz < 211 onto the w - plane with

, our original ... In order to obtain information on the

now use Lemma 3 . We know by Lemma 2 that all but a finite number of roots of ...

0 3 Rz < 211 onto the w - plane with

**zero**removed . Since sin z = ( 1/21 ) h ( eta ), our original ... In order to obtain information on the

**zeros**of M ( u ) from this , wenow use Lemma 3 . We know by Lemma 2 that all but a finite number of roots of ...

Page 2462

Moreover , if C belongs to the trace class 61 , then TnC converges to

norm , and CT * converges to

* / $ 1 } ) is conditionally compact , and thus for each ε > 0 there exists a finite ...

Moreover , if C belongs to the trace class 61 , then TnC converges to

**zero**in tracenorm , and CT * converges to

**zero**in trace norm . Proof . The set K = C ( { x € H ||* / $ 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 | |

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