## Linear Operators: Spectral theory |

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Page 909

Spectral Representation Let u be a finite positive measure defined on the

sets B of the complex plane and vanishing on the complement of a bounded set

S. One of the simplest examples of a bounded normal operator is the operator T ...

Spectral Representation Let u be a finite positive measure defined on the

**Borel**sets B of the complex plane and vanishing on the complement of a bounded set

S. One of the simplest examples of a bounded normal operator is the operator T ...

Page 913

We can clearly suppose that \ yol 1. Let Yo , Yı , Y2 , be an orthonormal basis for

H , whose initial element is yo . Let E be the spectral resolution for T and let vn ( e

) = ( E ( elyn , yn ) for each

We can clearly suppose that \ yol 1. Let Yo , Yı , Y2 , be an orthonormal basis for

H , whose initial element is yo . Let E be the spectral resolution for T and let vn ( e

) = ( E ( elyn , yn ) for each

**Borel**set e . Using the Lebesgue decomposition ...Page 1900

... ( 44 ) Boolean ring , definition , ( 40 ) representation of , 1.12.1 ( 41 )

of sets , definition , III.5.10 ( 137 )

function , X.I ( 891 )

of ...

... ( 44 ) Boolean ring , definition , ( 40 ) representation of , 1.12.1 ( 41 )

**Borel**fieldof sets , definition , III.5.10 ( 137 )

**Borel**function , X.1 ( 891 )**Borel**measurablefunction , X.I ( 891 )

**Borel**measure ( or**Borel**- Lebesgue measure ) , constructionof ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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