## Linear Operators: Spectral operators |

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Page 1953

If T has a closed

cases depending on whether the projection E ( { 0 } ) = 0 or not . First suppose

that E ( { 0 } ) = 0 . Then since the

...

If T has a closed

**range**, so does S . PROOF . The proof will be divided into twocases depending on whether the projection E ( { 0 } ) = 0 or not . First suppose

that E ( { 0 } ) = 0 . Then since the

**range**of T is closed , it follows from Corollary 12...

Page 1954

Q . E . D . It was shown in the course of the preceding proof that for an operator

Twith a closed

( { 0 } ' ) X . Thus for all sufficiently small complex numbers 1 + 0 the operator XI ...

Q . E . D . It was shown in the course of the preceding proof that for an operator

Twith a closed

**range**the point . = 0 is not in the spectrum of the operator V = T | E( { 0 } ' ) X . Thus for all sufficiently small complex numbers 1 + 0 the operator XI ...

Page 2312

The closure of the

such that y * x = 0 whenever T * y * = 0 . PROOF . If T * y * = 0 , then y * y = y * Tz =

( T * y * ) 2 = 0 for all y = Tz in the

...

The closure of the

**range**of a densely defined linear operator T is the set of all xsuch that y * x = 0 whenever T * y * = 0 . PROOF . If T * y * = 0 , then y * y = y * Tz =

( T * y * ) 2 = 0 for all y = Tz in the

**range**of T , and hence for all y in the closure of...

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