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

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Page 2419

... taken from the theory of singular integrals . 11 DEFINITION . Let y and ß be two

real numbers , with y 20 and 1 > ß > 0 . Let X be a B - space . The symbol A is

used for an X - valued function defined on the

... taken from the theory of singular integrals . 11 DEFINITION . Let y and ß be two

real numbers , with y 20 and 1 > ß > 0 . Let X be a B - space . The symbol A is

used for an X - valued function defined on the

**real axis**and we put ( 2 ) | | A | | 4 .Page 2424

6 ( B ( H ) ) , and let f be an infinitely differentiable H - valued function of a real

variable which vanishes outside a bounded interval of the

improper integral ( 28 ) ( ( A ) f ) ( 8 ) = P ] aveva nt A ( 8 , 0 ) f ( o ) . . 18 " exists .

6 ( B ( H ) ) , and let f be an infinitely differentiable H - valued function of a real

variable which vanishes outside a bounded interval of the

**real axis**. Then theimproper integral ( 28 ) ( ( A ) f ) ( 8 ) = P ] aveva nt A ( 8 , 0 ) f ( o ) . . 18 " exists .

Page 2456

We therefore find that equation ( 4 ) is valid whenever F is a continuous function

vanishing outside a compact subset of the

characteristic function of an interval of the

follows that ...

We therefore find that equation ( 4 ) is valid whenever F is a continuous function

vanishing outside a compact subset of the

**real axis**. Next let F be thecharacteristic function of an interval of the

**real axis**. Using Theorem XII . 2 . 6 , itfollows that ...

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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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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