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

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Page 1976

JEB SES Then for every bounded Borel scalar function y defined on the spectrum

o ( A ) , the

E - measurable function of s . The

JEB SES Then for every bounded Borel scalar function y defined on the spectrum

o ( A ) , the

**integral**( ii ) o ( a ) E ( d ) ; Â ( s ) ) O ( A ) is an e - essentially boundedE - measurable function of s . The

**integral**( iii ) Elo ; Â ( s ) ) e ( ds ) , 0EB , is a ...Page 1990

Here , we shall first be concerned with certain special examples of convolutions

which map H into H , which belong to the algebra A , and which have an

representation in one of the two forms ( 18 ) ( f * Q ) ( p ( s – t ) f ( t ) dt , QEH , RN

...

Here , we shall first be concerned with certain special examples of convolutions

which map H into H , which belong to the algebra A , and which have an

**integral**representation in one of the two forms ( 18 ) ( f * Q ) ( p ( s – t ) f ( t ) dt , QEH , RN

...

Page 2405

If he L ; ( S , E , p ) and 2 sr < 00 , it follows that the

) | h ( ) u ( dt ) ( 14 ) exists for u - almost all s , and that , writing fl . for the norm of

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 of

an 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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### 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 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