## Linear Operators: Spectral theory |

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Page 893

Returning now to the general integral s t ( s ) E ( ds ) where E is merely a

bounded

been defined in terms of the uniform operator topology . It is clear that if v is a

bounded ...

Returning now to the general integral s t ( s ) E ( ds ) where E is merely a

bounded

**additive**operator valued set function , we observe that the integral hasbeen defined in terms of the uniform operator topology . It is clear that if v is a

bounded ...

Page 932

Sz . - Nagy ' s original proof depended on the following theorem due to Neumark [

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

of S . Let F be an

Sz . - Nagy ' s original proof depended on the following theorem due to Neumark [

3 ] . THEOREM . 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 ...Page 958

Hence if e , and ex are disjoint then yle , u ez ) = Ele ; u ez ) yle u ez ) = [ E ( e ) +

Elez ) ] y ( e , u ez ) = E ( e ) yle , u ez ) + E ( en ) y ( e , vez ) = ylei ) + v ( ez ) , so

that the vector valued set function y is

...

Hence if e , and ex are disjoint then yle , u ez ) = Ele ; u ez ) yle u ez ) = [ E ( e ) +

Elez ) ] y ( e , u ez ) = E ( e ) yle , u ez ) + E ( en ) y ( e , vez ) = ylei ) + v ( ez ) , so

that the vector valued set function y is

**additive**on By . Therefore , if en ég = $ , the...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 other sections not shown

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero