## 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 2105

Berkson ( 2 ) showed that if E is a bounded spectral measure and if one defines |

| 3 | | = sup { var æ * E ( - ) w | 12 * 1 = 1 } , then | | · | | is a norm

and relative to which all the operators E ( 8 ) become Hermitian . It follows from ...

Berkson ( 2 ) showed that if E is a bounded spectral measure and if one defines |

| 3 | | = sup { var æ * E ( - ) w | 12 * 1 = 1 } , then | | · | | is a norm

**equivalent**to 1 : 1and relative to which all the operators E ( 8 ) become Hermitian . It follows from ...

Page 2115

It is proved that if T is decomposable and T and U are quasi - nilpotent

, then U is decomposable . Moreover , if T and U are decomposable , then X ( F )

= xy ( F ) for all closed sets F if and only if T and U are quasi - nilpotent ...

It is proved that if T is decomposable and T and U are quasi - nilpotent

**equivalent**, then U is decomposable . Moreover , if T and U are decomposable , then X ( F )

= xy ( F ) for all closed sets F if and only if T and U are quasi - nilpotent ...

Page 2320

... is evident that by subtracting a suitable multiple of B from C , we can pass to an

= 0 , ČIf ) = 0 , D ( f ) = 0 , B and 7 having order at most

...

... is evident that by subtracting a suitable multiple of B from C , we can pass to an

**equivalent**set of boundary conditions B ( f ) ... we can find an**equivalent**set B ( f )= 0 , ČIf ) = 0 , D ( f ) = 0 , B and 7 having order at most

**equal**to m , and Ď having...

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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