## 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 2070

Theorem 1 shows that the operator a determines a and f uniquely , and so we

may define a

that A , is a B - space under this

...

Theorem 1 shows that the operator a determines a and f uniquely , and so we

may define a

**norm**in A , by the equation ( 18 ) lalo = 10 + 1flo , ae A . . It is clearthat A , is a B - space under this

**norm**. It is also an algebra , for the product of two...

Page 2450

It follows that { Anx } is a Cauchy sequence of vectors for each x € X . Moreover ,

by definition of the

An , then , since R is a closed operator , we have Ax e D ( R ) for each x e X , and

...

It follows that { Anx } is a Cauchy sequence of vectors for each x € X . Moreover ,

by definition of the

**norm**in A , { RAnx } is also a Cauchy sequence . If A = limno .An , then , since R is a closed operator , we have Ax e D ( R ) for each x e X , and

...

Page 2462

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

trace

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 in

trace

**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