## Linear Operators: Spectral operators |

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Page 2152

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert Gardner Bartle. To

X , 0 ( x ) 58 } . It will be shown that M ( S ) is closed . For every x we have o ( x ) ...

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert Gardner Bartle. To

**prove**( C ) , let & be a closed subset of the complex plane and let M ( S ) = { w X eX , 0 ( x ) 58 } . It will be shown that M ( S ) is closed . For every x we have o ( x ) ...

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

Xin = x , then , by what we have already

where y , e Lac ( H ) and Y2 , Yz are orthogonal to ac ( H ) . But , since ... Using

this last fact , it is easy to

...

Xin = x , then , by what we have already

**proved**, we may write x = yı + y2 + Ys ,where y , e Lac ( H ) and Y2 , Yz are orthogonal to ac ( 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 | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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adjoint operator analytic applications arbitrary assumed B-space Banach space belongs Boolean algebra Borel sets boundary bounded bounded operator Chapter clear closed commuting compact complex condition consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined 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 perturbation plane positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero