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

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Page 2357

2 , each of these finite sums has a finite dimensional range and is hence

, it follows from Lemma V1 . 5 . 3 that for v > 0 the operator ( T - 101 ) - v is

+ do ...

2 , each of these finite sums has a finite dimensional range and is hence

**compact**, it follows from Lemma V1 . 5 . 3 that for v > 0 the operator ( T - 101 ) - v is

**compact**. Thus , if v > 0 , then since P + T = ( P + 101 ) + ( T - 107 ) , and since ( P+ do ...

Page 2360

It will also be shown that T - v is

, it will follow that B ( u ) = R ( u ; T + P ) is

large , so that the theorem will be proved . Let u be in V . To show that \ T ' ' R ( u ...

It will also be shown that T - v is

**compact**. From this , ( iii ) , and Theorem VI . 5 . 4, it will follow that B ( u ) = R ( u ; T + P ) is

**compact**for u in V , and i sufficientlylarge , so that the theorem will be proved . Let u be in V . To show that \ T ' ' R ( u ...

Page 2462

The operator C is

is complete . Q . E . D . 12 LEMMA . If C is a

uniformly bounded sequence of operators in H converging strongly to zero ...

The operator C is

**compact**by Corollary V1 . 5 . 5 , and thus proof of Corollary 11is complete . Q . E . D . 12 LEMMA . If C is a

**compact**operator in H , and { Tn } is auniformly bounded sequence of operators in H converging strongly to zero ...

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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