## Linear Operators: Spectral theory |

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Page 1017

calculate the trace of A relative to the

= ajy X ; = C - ais yj , j = 1 and so , CAC - 44 : = { : 3Y ; 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 AC - 1 yi = Ax i= ajy X ; = C - ais yj , j = 1 and so , CAC - 44 : = { : 3Y ; j = 1 From this it follows that

the trace of CAC - 1 , calculated relative to the

**basis**{ 91 , . . , yn } , is 2 - 1 Qii .Page 1029

Then , since S is necessarily invariant under T , there exists by the inductive

hypothesis , an orthonormal

0 for i > i . Let Xn be orthogonal to S and have norm one so that { x1 , . . . , xn } is

an ...

Then , since S is necessarily invariant under T , there exists by the inductive

hypothesis , an orthonormal

**basis**{ 21 , . . . , Xn - 1 } for S with ( ( T - ÎI ) Xi , x ; ) =0 for i > i . Let Xn be orthogonal to S and have norm one so that { x1 , . . . , xn } is

an ...

Page 1489

E _ ( a ) = I . Let v1 , . . . , Vk be a

for i = k + 1 , . . . , n . By the Hahn - Banach theorem , there exist functionals u , . . .

E _ ( a ) = I . Let v1 , . . . , Vk be a

**basis**for E + ( 22 ) E ” , and Vx + 1 , . . . , I ' m a**basis**for E _ ( 22 ) E " . Put vi ( 2 ) = E ( ) v ; for i = 1 , . . . , k , v ; ( 2 ) = E _ ( 7 ) v ;for i = k + 1 , . . . , n . By the Hahn - Banach theorem , there exist functionals u , . . .

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero