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

### From inside the book

Results 1-3 of 5

Page 2037

... 0 + 1 h which shows that Â ( s ) * ( 8 ) is

Ây . Thus Â 2 B . Now every positive , is in plÂ ) and thus the Hille - Yosida

theorem ( VIII . 1 . 13 ) shows that all large real 1 are in p ( Ân plẾ ) . For such , we

have ...

... 0 + 1 h which shows that Â ( s ) * ( 8 ) is

**square integrable**on RN and that By =Ây . Thus Â 2 B . Now every positive , is in plÂ ) and thus the Hille - Yosida

theorem ( VIII . 1 . 13 ) shows that all large real 1 are in p ( Ân plẾ ) . For such , we

have ...

Page 2042

... ( F¢® ) ( 8 ) ds = 4 ( 8 ) ( – 1 ) ( is ) * 450 % ( 8 ) ds . RN a RN Since 419 )

satisfies ( 79 ) , it follows that the coefficient of y ( s ) in the preceding integrand is

a1 ( 2 . ) ...

... ( F¢® ) ( 8 ) ds = 4 ( 8 ) ( – 1 ) ( is ) * 450 % ( 8 ) ds . RN a RN Since 419 )

satisfies ( 79 ) , it follows that the coefficient of y ( s ) in the preceding integrand is

**square integrable**on RN and the same is true of its Fourier transform F ( ( - 1 ) |a1 ( 2 . ) ...

Page 2043

... is continuous on RN and

21 THEOREM . Using the notation of Theorem 19 and letting B be an arbitrary

bounded linear operator in HP , we have : ( i ) The operator As + B with domain D

...

... is continuous on RN and

**square integrable**on RN , then q belongs to D ( A ) .21 THEOREM . Using the notation of Theorem 19 and letting B be an arbitrary

bounded linear operator in HP , we have : ( i ) The operator As + B with domain D

...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

### Other editions - View all

### 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