## Linear Operators: Spectral operators |

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Page 2149

If the resolvent set is

extensions of R ( ) ; T ' ) x must coincide on their common domain of continuity . Q

. E . D . All of the special type operators to be considered in the present section

will have ...

If the resolvent set is

**dense**, then any two analytic , or even continuous ,extensions of R ( ) ; T ' ) x must coincide on their common domain of continuity . Q

. E . D . All of the special type operators to be considered in the present section

will have ...

Page 2156

are both

^ , I – T ) Nx = 0 } is

) Nx = 0 } + { x | ( 121 – T ) ^ x = 0 } is also

are both

**dense**in X . Since M , is**dense**in X , the manifold ( 141 – T ) NM , + { 2 \ (^ , I – T ) Nx = 0 } is

**dense**in X , so that ( 1 , 1 – T ' ) ( 221 – T ) NX + { 2 | ( 9 , 1 – T) Nx = 0 } + { x | ( 121 – T ) ^ x = 0 } is also

**dense**in X . By Lemma 7 , 0 ( x ) < y ...Page 2159

The union of all intervals of constancy relative to T is an open set

PROOF . It is clear that the union of intervals of constancy is open . To see that it

is

do ...

The union of all intervals of constancy relative to T is an open set

**dense**in lo .PROOF . It is clear that the union of intervals of constancy is open . To see that it

is

**dense**, let y be a closed subarc of T ' , having positive length and let Yn = { 1odo ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

31 other sections not shown

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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