## 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 ( S ) is closed .

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

Theodore Schwartz. To

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

Page 2236

This

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

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

3.10 ) ...

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 TE ( en ) X is bounded ,

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

3.10 ) ...

Page 2459

If xn € Zac ( H ) and lim - Xn = x , then , by what we have already

write x = yı + y2 + Ys , where yı € Lac ( H ) and Y2 , Yz are orthogonal to Lac ( H ) .

... Using this last fact , it is easy to

If xn € Zac ( H ) and lim - Xn = x , then , by what we have already

**proved**, we maywrite x = yı + y2 + Ys , where yı € Lac ( H ) and Y2 , Yz are orthogonal to Lac ( H ) .

... Using this last fact , it is easy to

**prove**assertion ( c ) of the present lemma .### What people are saying - Write a review

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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