## Linear Operators: Spectral operators |

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Page 1994

Let any

Theorem II . ... Then , by Theorem 4 , ang = F - in X , and since ano

we see from the continuity of F that lims | | ( 8 ) — Am ( s ) / 2 ds = 0 , m . n + 00

DO which ...

Let any

**converge**for each o in H . Theorem 4 shows that Jan = lanco , andTheorem II . ... Then , by Theorem 4 , ang = F - in X , and since ano

**converges**,we see from the continuity of F that lims | | ( 8 ) — Am ( s ) / 2 ds = 0 , m . n + 00

DO which ...

Page 2218

It must be shown that { Ea }

so a consideration of the sequence { Ea – E } shows that it may be assumed that

E = 0 . Thus , to make an indirect proof , it is assumed that the sequence { Eq } is ...

It must be shown that { Ea }

**converges**strongly to E . By Lemma 6 , E is in B andso a consideration of the sequence { Ea – E } shows that it may be assumed that

E = 0 . Thus , to make an indirect proof , it is assumed that the sequence { Eq } is ...

Page 2319

Then , if f is in the domain D ( T ) , the series expansion ; T ) f

unconditionally in the topology of A ( n ) ( J ) . PROOF . The series 1 E ( Ni ; T ) f

certainly

, so ...

Then , if f is in the domain D ( T ) , the series expansion ; T ) f

**converges**to funconditionally in the topology of A ( n ) ( J ) . PROOF . The series 1 E ( Ni ; T ) f

certainly

**converges**unconditionally in the topology of L2 ( J ) . On the other hand, so ...

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