## Linear Operators: Spectral theory |

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Page 889

5 ) , every set in the domain of a spectral measure satisfying ( iii ) is necessarily

an open and closed subset of o ( T ) and ... a spectral measure which is defined

on the Boolean algebra B of all

for ...

5 ) , every set in the domain of a spectral measure satisfying ( iii ) is necessarily

an open and closed subset of o ( T ) and ... a spectral measure which is defined

on the Boolean algebra B of all

**Borel sets**in the plane and which satisfies ( iv )for ...

Page 894

2 ) that x * E ( 8 ) x = 0 for every

* € X * . It follows ( II . 3 . 15 ) that E ( d ) = 0 . Thus if E and A are bounded additive

regular operator valued set functions defined on the

2 ) that x * E ( 8 ) x = 0 for every

**Borel set**8 in S and every pair x , x * with x e X , x* € X * . It follows ( II . 3 . 15 ) that E ( d ) = 0 . Thus if E and A are bounded additive

regular operator valued set functions defined on the

**Borel sets**of a normal ...Page 913

Let E be the spectral resolution for T and let vn ( e ) = ( E ( e ) yn , yn ) for each

= 0 , and such that if e is a Borel subset of the complement en of en and no v ; ( e

) ...

Let E be the spectral resolution for T and let vn ( e ) = ( E ( e ) yn , yn ) for each

**Borel set**e . ... 14 ) , let { en } be a sequence of**Borel sets**such that n - ö v ; ( en )= 0 , and such that if e is a Borel subset of the complement en of en and no v ; ( e

) ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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 remark 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