## Linear Operators: Spectral operators |

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Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. n + 00 n

→ n + 00 Since f xe is a bounded function , the operator T ( f Xe ) is a bounded

operator . If x is in Esē ) X as well as in E ( e ) X , it

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. n + 00 n

→ n + 00 Since f xe is a bounded function , the operator T ( f Xe ) is a bounded

operator . If x is in Esē ) X as well as in E ( e ) X , it

**follows**from the operational ...Page 2246

it

domain of C . It is clear then that ( AI – C ) R ( 2 ) ąc = x for x in H and R ( a ) ( a1

— C ) x = x for x in D ( C ) , so that R ( a ) = R ( A ; C ) and 1 € ( C ) . On the other

hand ...

it

**follows**that R ( a ) is a bounded operator whose range is contained in thedomain of C . It is clear then that ( AI – C ) R ( 2 ) ąc = x for x in H and R ( a ) ( a1

— C ) x = x for x in D ( C ) , so that R ( a ) = R ( A ; C ) and 1 € ( C ) . On the other

hand ...

Page 2308

From this , it

. 1 . 5 , that ( 8 – XI ) - 1 is closed . Since this latter operator is everywhere defined

, it

From this , it

**follows**immediately that S - XI is closed , and hence from Lemma XII. 1 . 5 , that ( 8 – XI ) - 1 is closed . Since this latter operator is everywhere defined

, it

**follows**by the closed graph theorem ( II . 2 . 4 ) that it is bounded .### What people are saying - Write a review

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