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

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Page 1990

This elementary observation suggests that it might be easier to give certain

convolutions an integral representation if we use the integral of

functions , and this we shall do . We begin by defining the convolution integral (

19 ) ...

This elementary observation suggests that it might be easier to give certain

convolutions an integral representation if we use the integral of

**vector**valuedfunctions , and this we shall do . We begin by defining the convolution integral (

19 ) ...

Page 2160

It will next be shown that the

) X . To see this it will , in view of Corollary II . 3 . 13 , suffice to show that x * ( x - y

) = 0 for every linear functional a * which vanishes on ( 101 – T ) X . If x * is ...

It will next be shown that the

**vector**x - y is in the closure of the manifold ( 101 – T ') X . To see this it will , in view of Corollary II . 3 . 13 , suffice to show that x * ( x - y

) = 0 for every linear functional a * which vanishes on ( 101 – T ) X . If x * is ...

Page 2279

... in X * and let be a functional whose carrier is in ( * . If Nn N ( * ) = ( 0 ) , then

there exists a

xo > 0 for each bounded non - negative Borel function f such that S * ( f ) * * + 0 .

... in X * and let be a functional whose carrier is in ( * . If Nn N ( * ) = ( 0 ) , then

there exists a

**vector**Xo € X such that ( 1 ) y * ( x ) = 0 for all y * € N , ( 2 ) S * ( f ) *xo > 0 for each bounded non - negative Borel function f such that S * ( f ) * * + 0 .

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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