## Linear Operators: Spectral operators |

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Page 2256

... operator if and only if ( a ) the family of projections E ( o ; T ' ) corresponding to

compact spectral sets of T is uniformly bounded , and ( b ) no non -

satisfies the equation E ( 0 ) X = 0 for every compact spectral set o of T . PROOF .

... operator if and only if ( a ) the family of projections E ( o ; T ' ) corresponding to

compact spectral sets of T is uniformly bounded , and ( b ) no non -

**zero**x in Xsatisfies the equation E ( 0 ) X = 0 for every compact spectral set o of T . PROOF .

Page 2292

is the idempotent function of T corresponding to the analytic function which is one

near do and

) projects X onto the space of generalized eigenvectors corresponding to do .

is the idempotent function of T corresponding to the analytic function which is one

near do and

**zero**elsewhere near the spectrum of T and near infinity , then E ( 10) projects X onto the space of generalized eigenvectors corresponding to do .

Page 2325

0 3 Rz < 21 onto the w - plane with

our original ... In order to obtain information on the

now use Lemma 3 . We know by Lemma 2 that all but a finite number of roots of ...

0 3 Rz < 21 onto the w - plane with

**zero**removed . Since sin z = ( 1 / 21 ) h ( eta ) ,our original ... In order to obtain information on the

**zeros**of M ( u ) from this , wenow use Lemma 3 . We know by Lemma 2 that all but a finite number of roots of ...

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