## Linear Operators, Part 2 |

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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

Let Yo , 41 , 42 , . . . be an orthonormal basis for H , whose initial element is yo .

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

be a ...

Let Yo , 41 , 42 , . . . be an orthonormal basis for H , whose initial element is yo .

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

**Borel**set e . Using the Lebesgue decomposition theorem ( III . 4 . 14 ) , let { en }be a ...

Page 1900

12 . 1 ( 41 )

891 )

Lebesgue measure ) , construction of , ( 139 ) , III . 13 . 8 ( 223 )

measure ...

12 . 1 ( 41 )

**Borel**field of sets , definition , III . 5 . 10 ( 137 )**Borel**function , X . 1 (891 )

**Borel**measurable function , X . I ( 891 )**Borel**measure ( or**Borel**-Lebesgue measure ) , construction of , ( 139 ) , III . 13 . 8 ( 223 )

**Borel**- Stieltjesmeasure ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive adjoint operator 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 derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function give 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 shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero