## 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

T ( u ) = ap MP + ap - 148-1 + ... tao , ( 13 ) 772 ( u ) = bp up + bp - 140-1 + + .

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

Theodore Schwartz. If Regularity

**Hypothesis**1 is satisfied , we may write ap +0 ,T ( u ) = ap MP + ap - 148-1 + ... tao , ( 13 ) 772 ( u ) = bp up + bp - 140-1 + + .

Page 2397

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

XVIII.2.34 are satisfied . In the notation of that ...

2.34 is satisfied by virtue of Corollaries 9 and 11.

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

**hypotheses**of TheoremXVIII.2.34 are satisfied . In the notation of that ...

**Hypothesis**( i ) of Theorem XVIII.2.34 is satisfied by virtue of Corollaries 9 and 11.

**Hypothesis**( ii ) has been ...Page 2401

Using

= 9 ( Al ) . Now , by

most M ( M + M2 ) -1 . Thus , by Lemma VII.3.4 , the equation ( 6 ) B - ( B , A ) = A ...

Using

**hypothesis**( c ) , we may write this last equation as ( 5 ) P ( B – 4 ( B , A , ) )= 9 ( Al ) . Now , by

**hypothesis**, the map B + 4 ( B , A1 ) of A → A has norm atmost M ( M + M2 ) -1 . Thus , by Lemma VII.3.4 , the equation ( 6 ) B - ( B , A ) = A ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator discrete domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero