## 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 2141

Thus we may pass without loss of generality from consideration of the sequence {

om } to consideration of the sequence { 0m - 0 } ; that is , we may and shall

by the ...

Thus we may pass without loss of generality from consideration of the sequence {

om } to consideration of the sequence { 0m - 0 } ; that is , we may and shall

**assume**without loss of generality that o is void . Since Elom ) = E ( Om o ( T ) ) ,by the ...

Page 2150

Since 0 ( x ) Şo ( T ) we have M ( S ) = M ( 80 ( T ) ) , and it may therefore be

disconnected , the closed set 8 is an intersection na de of spectral sets da . Now

clearly M ( 8 ) ...

Since 0 ( x ) Şo ( T ) we have M ( S ) = M ( 80 ( T ) ) , and it may therefore be

**assumed**, without loss of generality , that 8 go ( T ) . Since o ( T ) is totallydisconnected , the closed set 8 is an intersection na de of spectral sets da . Now

clearly M ( 8 ) ...

Page 2212

To prove ( ix ) it may be

follows from ( vii ) that 0 , = 0 , 02 = ' w and ... that is , we shall

functions fz and fy are not identically equal and one of them , say fa , differs from

both fy ...

To prove ( ix ) it may be

**assumed**that on = 0 , , for if z = E , & and w = Ezy itfollows from ( vii ) that 0 , = 0 , 02 = ' w and ... that is , we shall

**assume**that thefunctions fz and fy are not identically equal and one of them , say fa , differs from

both fy ...

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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 formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm 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