## Linear Operators: Spectral operators |

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Page 2149

If the resolvent set is

extensions of R ( A ; 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 ...

If the resolvent set is

**dense**, then any two analytic , or even continuous ,extensions of R ( A ; 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 ...

Page 2156

are both

( 1 , 1 – T ) ^ x = 0 } is

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

are both

**dense**in X . Since M , is**dense**in X , the manifold ( 141 – T ' ) NM2 + { x |( 1 , 1 – T ) ^ x = 0 } is

**dense**in X , so that ( 141 – T ' ) N ( 12I – T ' ) NX + { x | ( 1 , 1– T ) Nx = 0 } + { x | ( 12I – T ) ^ x = 0 } is also

**dense**in X . By Lemma 7 , o ( x ) ...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

10 ...

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 T ' , having positive length and let Yn = { 1010 ...

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