## Linear Operators: Spectral theory |

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Page 889

As will be seen in the next section , a normal operator T in Hilbert space H

determines a spectral measure which is defined on the Boolean algebra B of all

measure ...

As will be seen in the next section , a normal operator T in Hilbert space H

determines a spectral measure which is defined on the Boolean algebra B of all

**Borel**sets in the plane and which satisfies ( iv ) for every de B . This spectralmeasure ...

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

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 } ...Page 1900

1 ( 41 )

B * - algebras , IX . 3 ( 874 – 879 ) . ( See also Algebra ) B - space ( or Banach

space ) , basic properties of , Chap . II definition , II . 3 . 2 . ( 59 ) integration , Chap

.

1 ( 41 )

**Borel**field of sets , definition , III . 5 . 10 20 B B - algebra . ( See Algebra )B * - algebras , IX . 3 ( 874 – 879 ) . ( See also Algebra ) B - space ( or Banach

space ) , basic properties of , Chap . II definition , II . 3 . 2 . ( 59 ) integration , Chap

.

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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