## Linear Operators: Spectral operators |

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Page 2154

Thus , since the

T – XZ ) " X $ ( T – AZ ) n + 1x . ... it follows that ( T – XI ) n + 1X + { x | ( T — ^ I ) n

+ 1x = 0 } is dense in X . By induction , it is seen that the

Thus , since the

**manifolds**( T – XI ) MX decrease as m increases , it follows that (T – XZ ) " X $ ( T – AZ ) n + 1x . ... it follows that ( T – XI ) n + 1X + { x | ( T — ^ I ) n

+ 1x = 0 } is dense in X . By induction , it is seen that the

**manifold**( T – X7 ] n + ...Page 2214

Then an operator is in the weakly closed algebra generated by B if and only if it

leaves invariant every closed linear

member of B . PROOF . It was observed in the preceding proof that an operator in

the ...

Then an operator is in the weakly closed algebra generated by B if and only if it

leaves invariant every closed linear

**manifold**which is invariant under everymember of B . PROOF . It was observed in the preceding proof that an operator in

the ...

Page 2351

Let X be a B - space , and T a discrete operator in X . Then by sp ( T ) , the

spectral span of T , we denote the smallest closed

sp ( T ) is ...

Let X be a B - space , and T a discrete operator in X . Then by sp ( T ) , the

spectral span of T , we denote the smallest closed

**manifold**containing all the**manifolds**E ( 1 ; T ) X with X in o ( T ) . REMARK . It follows , by Lemma 2 . 2 , thatsp ( T ) is ...

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