## Linear Operators: Spectral theory |

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Page 932

Let S be an abstract set and E a field ( resp . o - field ) of subsets of S . Let F be an

operators on a Hilbert space H satisfying F ( 0 ) = 0 and F ( S ) = 1 . Then there

exists ...

Let S be an abstract set and E a field ( resp . o - field ) of subsets of S . Let F be an

**additive**( resp . weakly countably**additive**) function on to the set of positiveoperators on a Hilbert space H satisfying F ( 0 ) = 0 and F ( S ) = 1 . Then there

exists ...

Page 958

Hence if e , and ez are disjoint then yle , ue , ) = E ( e , u ez ) y ( e , u ez ) = [ E ( ez

) + E ( ez ) ] y ( e , u ez ) = E ( en ) y ( e , u ez ) + E ( ez ) y ( e , u ez ) = ylei ) + ylez )

, so that the vector valued set function y is

Hence if e , and ez are disjoint then yle , ue , ) = E ( e , u ez ) y ( e , u ez ) = [ E ( ez

) + E ( ez ) ] y ( e , u ez ) = E ( en ) y ( e , u ez ) + E ( ez ) y ( e , u ez ) = ylei ) + ylez )

, so that the vector valued set function y is

**additive**on B . Therefore , if ein ez ...Page 1803

Solid analytical geometry and determinants . H . Holt Co . , New York , 1930 .

Dubrovski ( Doubrovsky ) , V . M . 1 . On some properties of completely

set functions and their application to generalization of a theorem of Lebesgue .

Mat .

Solid analytical geometry and determinants . H . Holt Co . , New York , 1930 .

Dubrovski ( Doubrovsky ) , V . M . 1 . On some properties of completely

**additive**set functions and their application to generalization of a theorem of Lebesgue .

Mat .

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive 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 differential equations 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 Proc 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