## Linear Operators: Spectral operators |

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Page 2025

Since Ex : ( s ) # 0 there is a vector f ( s ) in EP with \ ¥ ( s ) = 1 and ( s ) = Êxj ( s ) (

s ) . Thus , since Êx ; ( s ) and Êxg ( s ) are disjoint projections if q # j , it follows

from ( 35 ) that R ( a ; Â ( s ) ) " } ( s ) = 4 ( s ) ( 1 - dxy ( s ) ) - - and ( 39 )

Axy ...

Since Ex : ( s ) # 0 there is a vector f ( s ) in EP with \ ¥ ( s ) = 1 and ( s ) = Êxj ( s ) (

s ) . Thus , since Êx ; ( s ) and Êxg ( s ) are disjoint projections if q # j , it follows

from ( 35 ) that R ( a ; Â ( s ) ) " } ( s ) = 4 ( s ) ( 1 - dxy ( s ) ) - - and ( 39 )

**gives**-Axy ...

Page 2031

To prove ( iv ) suppose first that ys is in 0 in which case a permissible interchange

of integration

ds = Trulm ) , de D . Now 0 is dense in H = L2 ( RN ) and thus there is for an ...

To prove ( iv ) suppose first that ys is in 0 in which case a permissible interchange

of integration

**gives**RN RN ( FT . ) q ) = S ( Fp ) ( $ ) { ( 8 ) ds = S 968 ) ( F % ) ( 3 )ds = Trulm ) , de D . Now 0 is dense in H = L2 ( RN ) and thus there is for an ...

Page 2065

Thus the Gelfand theory

not an element possesses an inverse and so ... In order to apply this procedure to

a given algebra , it is sufficient to

...

Thus the Gelfand theory

**gives**a general procedure for determining whether ornot an element possesses an inverse and so ... In order to apply this procedure to

a given algebra , it is sufficient to

**give**a satisfactory representation of its spectrum...

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