## Linear Operators: Spectral theory |

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Page 1017

calculate the trace of A relative to the

20 ; 28 ; = C + + ] : 3 Ys , j = 1 and so , CAC - 4y = { ( , 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 - 14 : = Ax =20 ; 28 ; = C + + ] : 3 Ys , j = 1 and so , CAC - 4y = { ( , j = 1 From this it follows that

the trace of CAC - 1 , calculated relative to the

**basis**{ Y1 , . . . , yn } , is n = 1 Qii ...Page 1029

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

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

orthonormal

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

Page 1489

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

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

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

**basis**for E + ( 24 ) E " , and Vx + 1 , . . . , V ' n a**basis**for E _ ( 21 ) E " . Put vi ( 2 ) = E ( 2 ) vi for i = 1 , . . . , k , v ( a ) = E _ ( a ) v ;for i = k + 1 , . . . , n . By the Hahn - Banach theorem , there exist functionals u , . . .

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

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 consider 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 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