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

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Page 2393

Moreover , since A + ( a ) # 0 , A - ( a ) + 0 , 0 < < 00 , Z has no limit points on the

of do as a pole of the resolvent . By Theorems VII . 3 . 18 and VII . 3 . 24 , each fe

E ...

Moreover , since A + ( a ) # 0 , A - ( a ) + 0 , 0 < < 00 , Z has no limit points on the

**real axis**. Consequently , Z is a finite set . Now let do € Z and let vo be the orderof do as a pole of the resolvent . By Theorems VII . 3 . 18 and VII . 3 . 24 , each fe

E ...

Page 2424

6 ( B ( H ) ) , and let f be an infinitely differentiable H - valued function of a real

variable which vanishes outside a bounded interval of the

improper integral pt A ( s , o ) f ( o ) do ( 28 ) ( T ( A ) f ) ( s ) = P J - S - O exists .

6 ( B ( H ) ) , and let f be an infinitely differentiable H - valued function of a real

variable which vanishes outside a bounded interval of the

**real axis**. Then theimproper integral pt A ( s , o ) f ( o ) do ( 28 ) ( T ( A ) f ) ( s ) = P J - S - O exists .

Page 2456

We therefore find that equation ( 4 ) is valid whenever F is a continuous function

vanishing outside a compact subset of the

characteristic function of an interval of the

follows that ...

We therefore find that equation ( 4 ) is valid whenever F is a continuous function

vanishing outside a compact subset of the

**real axis**. Next let F be thecharacteristic function of an interval of the

**real axis**. Using Theorem XII . 2 . 6 , itfollows that ...

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