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

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Page 2174

In a Hilbert space the condition that lettr | SM for all te R implies that T is

equivalent to a self adjoint operator and

spectrum . ( This follows from Lemma XV . 6 . 1 which implies that the bounded

group G ...

In a Hilbert space the condition that lettr | SM for all te R implies that T is

equivalent to a self adjoint operator and

**hence**is a scalar type operator with realspectrum . ( This follows from Lemma XV . 6 . 1 which implies that the bounded

group G ...

Page 2308

17 and the remark preceding Definition XIII . 2 . 29 , that S is closed . From this , it

follows immediately that S - XI is closed , and

8 – 17 ) - 1 is closed . Since this latter operator is everywhere defined , it follows ...

17 and the remark preceding Definition XIII . 2 . 29 , that S is closed . From this , it

follows immediately that S - XI is closed , and

**hence**from Lemma XII . 1 . 5 , that (8 – 17 ) - 1 is closed . Since this latter operator is everywhere defined , it follows ...

Page 2357

then it is clear that L is a bounded operator and that ( S – XI ) - = ( T - XI ) - ' L .

( S – XI ) - v is a bounded operator which is compact if P ( T – XI ) - " is compact (

cf .

then it is clear that L is a bounded operator and that ( S – XI ) - = ( T - XI ) - ' L .

**Hence**( P + N ) ( S – XI ) - v = P ( S – XI ) - ° + N ( S — 21 ) - " = P ( T – XI ) - L + N( S – XI ) - v is a bounded operator which is compact if P ( T – XI ) - " is compact (

cf .

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