## Linear Operators: Spectral theory |

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Page 890

2x } of T and otherwise let E ( 8 ) be the sum of all the projections E ( 2x ) for

which Ried , then the function E is a resolution of the identity for T and the

operational calculus is given by the

where the ...

2x } of T and otherwise let E ( 8 ) be the sum of all the projections E ( 2x ) for

which Ried , then the function E is a resolution of the identity for T and the

operational calculus is given by the

**formula**( vi ) ( T ) = Soxxy : ( 2 ) E ( 11 ) ,where the ...

Page 1089

This

, ) = ... ( P.Pn ) = 0 Since 1T9 | 2 = ( T9 , Tp ) ( T * T9,9 ) , we see our lemma to be

a special case of the “ minimax

This

**formula**may be written ( un ( T ) ) 2 = min 91 , ... , Pin max 17912 19 = 1 ( 4,8, ) = ... ( P.Pn ) = 0 Since 1T9 | 2 = ( T9 , Tp ) ( T * T9,9 ) , we see our lemma to be

a special case of the “ minimax

**formula**” for the eigenvalues of a compact ...Page 1363

basis for this

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

calculated from the resolvent by the

; T ) -R ...

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 ( 12 , ) may be

calculated from the resolvent by the

**formula**1 pla El ( 27 , ) ) } = - lim lim [ R ( 1 - is; T ) -R ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

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 Russian satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero