## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2107

To cite one result , Apostol [ 6 ] proved that if f is analytic on a neighborhood of o (

T ) , if f ' ( a ) # 0 for de o ( T ) , and if f ( T ) is a spectral (

operator , then T is spectral (

To cite one result , Apostol [ 6 ] proved that if f is analytic on a neighborhood of o (

T ) , if f ' ( a ) # 0 for de o ( T ) , and if f ( T ) is a spectral (

**respectively**, scalar type )operator , then T is spectral (

**respectively**, scalar type ) . A number of papers ...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 ...

( 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 ...

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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 formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal 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