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

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Page 2152

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

Theodore Schwartz. To

and let M ( 8 ) = { x | x € X , ( x ) $ 8 } . It will be shown that M ( 8 ) is closed .

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

Theodore Schwartz. To

**prove**( C ) , let & be a closed subset of the complex planeand let M ( 8 ) = { x | x € X , ( x ) $ 8 } . It will be shown that M ( 8 ) is closed .

Page 2236

This

) + g ( T ) ) and let { en } be as above . Then , since T | E ( en ) X is bounded ,

statements ( i ) and ( ii ) and the functional calculus of bounded operators ( cf . VII

.

This

**proves**( iii ) . The proof of ( viii ) is evident . To**prove**( vi ) , let x be in D ( f ( T) + g ( T ) ) and let { en } be as above . Then , since T | E ( en ) X is bounded ,

statements ( i ) and ( ii ) and the functional calculus of bounded operators ( cf . VII

.

Page 2459

Xn = x , then , by what we have already

where yı € Lac ( H ) and Y2 , Yz are orthogonal to Eac ( H ) . But , since ... Using

this last fact , it is easy to

...

Xn = x , then , by what we have already

**proved**, we may write x = yı + y2 + Yu ,where yı € Lac ( H ) and Y2 , Yz are orthogonal to Eac ( H ) . But , since ... Using

this last fact , it is easy to

**prove**assertion ( c ) of the present lemma . We argue as...

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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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### 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