## 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 | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero