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

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Page 2245

Unfortunately , no very literal extension can be given , as is shown by the

following example of a spectral operator T such that sin aT is not a spectral

operator . For n 2 1 , let Hinn be an n - dimensional unitary space with

Unfortunately , no very literal extension can be given , as is shown by the

following example of a spectral operator T such that sin aT is not a spectral

operator . For n 2 1 , let Hinn be an n - dimensional unitary space with

**orthonormal**basis ( 2n ) ...Page 2425

Let { Xq , & € A } be a complete

va = ( Him ( * + L•396 – o , po do , wa ) = link + LJplo - os = * * 06 . 0 ) ( 5 ( 0 ) , wa

) do = g * * ( f ( ) , wa ) do – p * * ( 1 m . p ( 8 – 0 ) ( S ( O ) , ta ) do . 1 - 0 8 - 0 ...

Let { Xq , & € A } be a complete

**orthonormal**basis for H . Then 8 + & 0 ( 38 ) ( gl ) ,va = ( Him ( * + L•396 – o , po do , wa ) = link + LJplo - os = * * 06 . 0 ) ( 5 ( 0 ) , wa

) do = g * * ( f ( ) , wa ) do – p * * ( 1 m . p ( 8 – 0 ) ( S ( O ) , ta ) do . 1 - 0 8 - 0 ...

Page 2451

... 2 ( x , xm ) xn n , m = 1 dm - dm mm for x e X and A A . By Schwarz ' inequality ,

and since { xn } is an

Xn ) 2 s : ( 8 7 14 . - 1 ' ) ( 3 = ) slalo . É 5 * | ( 48m , talle m = 1 # n m = 1 Inim n ...

... 2 ( x , xm ) xn n , m = 1 dm - dm mm for x e X and A A . By Schwarz ' inequality ,

and since { xn } is an

**orthonormal**basis , we have n = 1 m = 1 men Ern ? | ( Axm ,Xn ) 2 s : ( 8 7 14 . - 1 ' ) ( 3 = ) slalo . É 5 * | ( 48m , talle m = 1 # n m = 1 Inim n ...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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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 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