## 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 ' ) ^ x = 0 } is

= 0 } + { x | ( 12 I – T ) Nx = 0 } is also

are both

**dense**in X. Since M , is**dense**in X , the manifold ( 141 – T ) NM , + { x | (I – T ' ) ^ x = 0 } is

**dense**in X , so that ( 1,1 – T ) ^ ( 1,1 – T ) NX + { x | ( 1,1 – T ) Nx= 0 } + { x | ( 12 I – T ) Nx = 0 } is also

**dense**in X. By Lemma 7 , ( x ) < y if ( 1,1 ...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

...

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

**dense**in To .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 I " , having positive length and let γη = { λολο...

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

32 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 countably additive defined Definition denote dense determined differential operator 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