## 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 2063

2 that if the elements a , A in A , A ” ,

1 , A - 1 existing as operators in H , HP ,

A , AP ,

2 that if the elements a , A in A , A ” ,

**respectively**, possess bounded inverses a -1 , A - 1 existing as operators in H , HP ,

**respectively**, then these inverses are inA , AP ,

**respectively**. The proof given in Lemma 9 . 2 for these statements used ...Page 2108

Let u (

in order for there to exist a necessarily unique spectral measure d on the Baire

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 . Thenin order for there to exist a necessarily unique spectral measure d on the Baire

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

...

Page 2109

( c ) The mappings u → - d u / dta + tau on the spaces ♡ (

rapidly decreasing c functions (

line are scalar . Thus the Fourier transform u ( t ) → SR e - 2nitsu ( s ) ds in 0 and

its ...

( c ) The mappings u → - d u / dta + tau on the spaces ♡ (

**respectively**, Tl ) ofrapidly decreasing c functions (

**respectively**, tempered distributions ) on the realline are scalar . Thus the Fourier transform u ( t ) → SR e - 2nitsu ( s ) ds in 0 and

its ...

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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