## 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 2108

Let u ( respectively , v ) be a spectral measure on X ( respectively , Y ) to A . Then

in order for there to

sets in X X Y to A such that ( 8 x 0 ) = u ( 8 ) v ( o ) for all 8 , o , it is necessary and

...

Let u ( respectively , v ) be a spectral measure on X ( respectively , Y ) to A . Then

in order for there to

**exist**a necessarily unique spectral measure d on the Bairesets in X X Y to A such that ( 8 x 0 ) = u ( 8 ) v ( o ) for all 8 , o , it is necessary and

...

Page 2278

There

for each F * € C * , m ( F * ) is the least cardinal power of a set of cyclic subspaces

spanning F * X * in the X - topology . There is a unique decomposition of the ...

There

**exists**a unique multiplicity function m defined on B * with the property thatfor each F * € C * , m ( F * ) is the least cardinal power of a set of cyclic subspaces

spanning F * X * in the X - topology . There is a unique decomposition of the ...

Page 2405

If he L ; ( S , E , p ) and 2 sr < 00 , it follows that the integral ( Ah ) ( 8 ) = S 14 ( 8 , t

) | h ( ) u ( dt ) ( 14 )

an element f of L ( S , E , u ) , we have Ahl , $ { A } , \ h \ , . Thus , using Theorem ...

If he L ; ( S , E , p ) and 2 sr < 00 , it follows that the integral ( Ah ) ( 8 ) = S 14 ( 8 , t

) | h ( ) u ( dt ) ( 14 )

**exists**for u - almost all s , and that , writing fl . for the norm ofan element f of L ( S , E , u ) , we have Ahl , $ { A } , \ h \ , . Thus , using Theorem ...

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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