## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 2152

To

To

**prove**( C ) , let 8 be a closed subset of the complex plane and let M ( 8 ) = { x | x € X , ( x ) $ 8 } . It will be shown that M ( S ) is closed . For every 3 we have o ( x ) Şo ( T ) = lo and thus M ( S ) = M ( ST . ) ...Page 2236

This

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 ...Page 2459

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

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

**proved**, we may write x = yı + y2 + Ys , where yı € Lac ( H ) and Y2 , Yz are ... 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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### Common terms and phrases

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