## 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 ( 0 ; 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 ( 0 ; T ' )

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

non -

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

Page 2325

In order to obtain information on the

Lemma 3 . ... Divide the rectangle 271 ZR ( 2 — ) 20 into two rectangles R ( 1 )

and R ( 2 ) , each of which contains exactly one

R ( 1 ) ...

In order to obtain information on the

**zeros**of M ( u ) from this , we now useLemma 3 . ... Divide the rectangle 271 ZR ( 2 — ) 20 into two rectangles R ( 1 )

and R ( 2 ) , each of which contains exactly one

**zero**of sin ( 2 — « ) = B . Put R1 =R ( 1 ) ...

Page 2462

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

trace norm , and CT * converges to

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 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