## Linear Operators: Spectral operators |

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there exists a complex number 1 € 0 ( T ) . It is clear that without any loss of

generality we may assume that i = 0 . Let R = T - 1 . Then , by Theorem VII . 9 .

**Suppose**first that ( a ) and ( b ) are satisfied . Since o ( T ) is totally disconnected ,there exists a complex number 1 € 0 ( T ) . It is clear that without any loss of

generality we may assume that i = 0 . Let R = T - 1 . Then , by Theorem VII . 9 .

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However , rather than seek the maximum generality , it will be convenient to

Both these properties hold for B if X is separable , so this will be assumed for the

rest of ...

However , rather than seek the maximum generality , it will be convenient to

**suppose**that B is itself complete and satisfies the countable chain condition .Both these properties hold for B if X is separable , so this will be assumed for the

rest of ...

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enumeration of its spectrum . Let dn denote the distance from an to o ( T ) — { an }

.

.

**Suppose**that E is its resolution of the identity , and**suppose**that { an } is anenumeration of its spectrum . Let dn denote the distance from an to o ( T ) — { an }

.

**Suppose**that for all but a finite number of n , Elan ) has a onedimensional range.

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