## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 1926

2EO ( T ) In terms of N and E the operational calculus for an

takes the form f ( T ) = ] | fem ( A ) E ( ) , do ( T ) whereas if T is normal only the first

term f ( T ) = L , f ( A ) E ( 21 ) O ( T ) of this series is needed to express f ( T ) .

2EO ( T ) In terms of N and E the operational calculus for an

**arbitrary**operatortakes the form f ( T ) = ] | fem ( A ) E ( ) , do ( T ) whereas if T is normal only the first

term f ( T ) = L , f ( A ) E ( 21 ) O ( T ) of this series is needed to express f ( T ) .

Page 2031

... in 0 in which case a permissible interchange of integration gives ( FT ) ) ( Q ) =

1 ( FP ) ( s ) ¥ ( s ) ds p ( s ) ( F4 ) ( s ) ds = Tfvlo ) , OEO . " RN RN ' Now 0 is

dense in H = L2 ( RN ) and thus there is for an

with ...

... in 0 in which case a permissible interchange of integration gives ( FT ) ) ( Q ) =

1 ( FP ) ( s ) ¥ ( s ) ds p ( s ) ( F4 ) ( s ) ds = Tfvlo ) , OEO . " RN RN ' Now 0 is

dense in H = L2 ( RN ) and thus there is for an

**arbitrary**4 in H a sequence { 4n } cwith ...

Page 2314

Let ko , ką be

) defined by the formal differential operator 7 = - ( d / dt ) 2 and the boundary

conditions f ( 0 ) – ko f ' ( 0 ) = 0 , $ ( 1 ) – kif ' ( 1 ) = 0 . Then T is a spectral

operator ...

Let ko , ką be

**arbitrary**constants and let T be the unbounded operator in L2 ( 0 , 1) defined by the formal differential operator 7 = - ( d / dt ) 2 and the boundary

conditions f ( 0 ) – ko f ' ( 0 ) = 0 , $ ( 1 ) – kif ' ( 1 ) = 0 . Then T is a spectral

operator ...

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