## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 890

... all the projections E ( 1 ) for which died , then the function E is a resolution of

the identity for T and the operational calculus is given by the

Sove , t ( 1 ) E ( da ) , where the integral is defined as the finite sum - 1 | ( ) E ( ) :) .

... all the projections E ( 1 ) for which died , then the function E is a resolution of

the identity for T and the operational calculus is given by the

**formula**( vi ) f ( T ) =Sove , t ( 1 ) E ( da ) , where the integral is defined as the finite sum - 1 | ( ) E ( ) :) .

Page 1112

Ann . bnn Therefore , by Lagrange's expansion

matrix inverses , we have d d i = 1 j = 1 Vii d det ( A + zB ) | , -0 E bij Vsi dz det ( A

) tr ( 4-1B ) , where denotes the cofactor of the element air of the matrix A.

Ann . bnn Therefore , by Lagrange's expansion

**formula**and Cramer's**formula**formatrix inverses , we have d d i = 1 j = 1 Vii d det ( A + zB ) | , -0 E bij Vsi dz det ( A

) tr ( 4-1B ) , where denotes the cofactor of the element air of the matrix A.

Page 1363

basis for this

projection in the resolution of the identity for T corresponding to ( 11 , 12 ) may be

calculated from the resolvent by the

R ( 1 ...

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 ( 11 , 12 ) may be

calculated from the resolvent by the

**formula**80E 0 + 2πι 1 E ( ( 2 , 2 ) ) = lim lim [R ( 1 ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

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### Common terms and phrases

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 Nauk 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