## Linear Operators: Spectral theory |

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Page 1017

calculate the trace of A relative to the

Σ α , α , = C- ' Σα ; 95 , j = 1 and so , n CAC- £ y ; = 241593 j = 1 From this it follows

that the trace of CAC - 1 , calculated relative to the

calculate the trace of A relative to the

**basis**Yı , ... , Yn . Note that n n AC - yi Αα , =Σ α , α , = C- ' Σα ; 95 , j = 1 and so , n CAC- £ y ; = 241593 j = 1 From this it follows

that the trace of CAC - 1 , calculated relative to the

**basis**{ 91 , ... , Yn } , is XI ...Page 1029

Let xn be orthogonal to S and have norm one so that { 81 , ... , xn } is an

orthonormal

- Î1 ) x ;, xz ) and has ( ( T- ÀI ) x ; , x ; ) = 0 for 1 > i . This completes the

construction of the ...

Let xn be orthogonal to S and have norm one so that { 81 , ... , xn } is an

orthonormal

**basis**for E " . Then the matrix of T - ÀI in terms of { x1 , ... , xn } is ( ( T- Î1 ) x ;, xz ) and has ( ( T- ÀI ) x ; , x ; ) = 0 for 1 > i . This completes the

construction of the ...

Page 1489

Let 01 , ... , Vx be a

. Put vi ( 2 ) = E ( 2 ) vi for i 1 , v ; ( 2 ) = E_ ( 2 ) v , for i = k + 1 , ... , n . By the Hahn

- Banach theorem , there exist functionals um , ... , u * ( E ) * such that u ?

Let 01 , ... , Vx be a

**basis**for E4 ( 14 ) E " , and Vx + 1 , ... , I'm a**basis**for E_ ( 2 ) E. Put vi ( 2 ) = E ( 2 ) vi for i 1 , v ; ( 2 ) = E_ ( 2 ) v , for i = k + 1 , ... , n . By the Hahn

- Banach theorem , there exist functionals um , ... , u * ( E ) * such that u ?

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