## Linear Operators: Spectral theory |

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Page 890

... the projections E ( 2 ; ) for which mied , then the function E is a resolution of the

identity for T and the operational calculus is given by the

t ( ) E ( da ) , where the integral is defined as the finite sum ku , ( 2 ; ) E ( 2x ) .

... the projections E ( 2 ; ) for which mied , then the function E is a resolution of the

identity for T and the operational calculus is given by the

**formula**( vi ) f ( T ) = fan ,t ( ) E ( da ) , where the integral is defined as the finite sum ku , ( 2 ; ) E ( 2x ) .

Page 1112

Ann bni bna . . . bnn Therefore , by Lagrange ' s expansion

s

= det ( 4 ) tr ( 4 - B ) , where Vis denotes the cofactor of the element ais of the ...

Ann bni bna . . . bnn Therefore , by Lagrange ' s expansion

**formula**and Cramer 's

**formula**for matrix inverses , we have dd det ( A + zB ) / 2 = 0 = 2 bij Vsi i = 1 j = 1= det ( 4 ) tr ( 4 - B ) , where Vis denotes the cofactor of the element ais of the ...

Page 1363

basis for this

projection in the resolution of the identity for T corresponding to ( ay , 22 ) may be

calculated from the resolvent by the

E 0 ...

basis for this

**formula**is found in Theorem XII . 2 . 10 which asserts that theprojection in the resolution of the identity for T corresponding to ( ay , 22 ) may be

calculated from the resolvent by the

**formula**pie - o E ( ( 27 , 2 , ) ) = lim lim = 8 - 0E 0 ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive Akad 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 determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f 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 singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero