## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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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 have ...

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 have ...

Page 2156

are both

1 – T ) ^ 2 = 0 } is

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

are both

**dense**in X . Since M , is**dense**in X , the manifold ( jI – T ) NM , + { x | ( 1 ,1 – T ) ^ 2 = 0 } is

**dense**in X , so that ( 141 – T ) ^ ( 12 I – T ) NX + { x | ( 1 , 1 – T )^ x = 0 } + { x | ( 121 – T ) Nx = 0 } is also

**dense**in X . By Lemma 7 , g ( x ) < y if ...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 I . .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 Yn = { 10 /10 ...

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