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

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Page 2323

Spectral Theory : Self Adjoint Operators in Hilbert Space Nelson Dunford, Jacob

Theodore Schwartz. If Regularity

= Apped + ap - 1MP - lt . . . tao , ap 70 , ( 13 ) 772 ( u ) = bp up + bp - 148 - 1 + .

Spectral Theory : Self Adjoint Operators in Hilbert Space Nelson Dunford, Jacob

Theodore Schwartz. If Regularity

**Hypothesis**1 is satisfied , we may write 77 ( u )= Apped + ap - 1MP - lt . . . tao , ap 70 , ( 13 ) 772 ( u ) = bp up + bp - 148 - 1 + .

Page 2397

The theorem will follow as soon as it is shown that the

XVIII . 2 . 34 are ...

7 ( iv ) . It therefore only remains to establish

.

The theorem will follow as soon as it is shown that the

**hypotheses**of TheoremXVIII . 2 . 34 are ...

**Hypothesis**( ii ) has been established and is given by Lemma7 ( iv ) . It therefore only remains to establish

**hypothesis**( iii ) of Theorem XVIII . 2.

Page 2401

Using

( A1 ) . Now , by

M2 ( M1 + M2 ) - 1 . Thus , by Lemma VII . 3 . 4 , the equation ( 6 ) B – 4 ( B , Aj ) ...

Using

**hypothesis**( c ) , we may write this last equation as q ( B – 4 ( B , A1 ) ) = Q( A1 ) . Now , by

**hypothesis**, the map B + 4 ( B , A1 ) of A → A has norm at mostM2 ( M1 + M2 ) - 1 . Thus , by Lemma VII . 3 . 4 , the equation ( 6 ) B – 4 ( B , Aj ) ...

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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