## 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 2381

If S is some other positive constant and S < € , then ag 2 ag . Let hu ( t ) , ( t , u ) € [

ag , 00 ) * Pi , be the

above . Then , by the uniqueness of the

If S is some other positive constant and S < € , then ag 2 ag . Let hu ( t ) , ( t , u ) € [

ag , 00 ) * Pi , be the

**solution**of equation ( 7 ) which exists in C [ ag , oo ) , by theabove . Then , by the uniqueness of the

**solution**of ( 7 ) , we have hu ( t ) = hult ) ...Page 2391

Appropriate choice of this second

properties of the resolvent as needed below . The following corollary summarizes

the necessary facts in a form convenient for later use . 6 COROLLARY . Let Aa )

and u ...

Appropriate choice of this second

**solution**will enable us to calculate finerproperties of the resolvent as needed below . The following corollary summarizes

the necessary facts in a form convenient for later use . 6 COROLLARY . Let Aa )

and u ...

Page 2394

there exists a

for all sufficiently small u e P + , such that oz and os are continuous in t and Me ...

**solutions**ởi = 0 ( t , - u ( a ) ) and ( i = 0 ; lt , ula ) ) of the equation to = do . ... Thenthere exists a

**solution**oz ( t , u ) of the equation to = u o , defined for 0 St < oo andfor all sufficiently small u e P + , such that oz and os are continuous in t and Me ...

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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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### Common terms and phrases

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