## Linear Operators, Part 2 |

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Page 1017

calculate the trace of A relative to the

AC - y = 48 . – Żages = c - 95 , CAC - y = žaves 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 j = 1 and so ,AC - y = 48 . – Żages = c - 95 , CAC - y = žaves j = 1 From this it follows that the

trace of CAC - 1 , calculated relative to the

**basis**{ 91 , . . . , Yn } , is { n - 1 Qiz .Page 1029

Let S be an n - 1 dimensional subspace of En such that S 2 Sg . Then , since S is

necessarily invariant under T , there exists by the inductive hypothesis , an

orthonormal

xn be ...

Let S be an n - 1 dimensional subspace of En such that S 2 Sg . Then , since S is

necessarily invariant under T , there exists by the inductive hypothesis , an

orthonormal

**basis**{ x1 , . . . , Xn - 1 } for S with ( ( T - ÀI ) x ; , x ; ) = 0 for i > i . Letxn be ...

Page 1489

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

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

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

**basis**for E + ( 12 ) E " , and Vx + 1 , . . . , 7 ' n a**basis**for E _ ( 12 ) E ” . Put vila ) = E4 ( 2 ) vi for i = 1 , . . . , k , v ; ( 2 ) = E _ ( a ) o ;for i = k + 1 , . . . , n . By the Hahn - Banach theorem , there exist functionals um ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function give given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem 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 vanishes vector zero