## Linear Operators: Spectral theory |

### From inside the book

Results 1-3 of 93

Page 898

If E is the resolution of the

of complex numbers , then E ( 8 ) T = TE ( 8 ) , ( Ts ) CJ , where To is the

restriction of T to E ( 8 ) H . Proof . The first statement follows from Theorem 1 ( ii )

.

If E is the resolution of the

**identity**for the normal operator T and if d is a Borel setof complex numbers , then E ( 8 ) T = TE ( 8 ) , ( Ts ) CJ , where To is the

restriction of T to E ( 8 ) H . Proof . The first statement follows from Theorem 1 ( ii )

.

Page 920

Under this assumption it will be shown that there is an ordered representation of

H onto , Lzlēn , ñ ) relative to T . It will follow from Theorem 10 that U and Ở are

equivalent . Let E and Ể be the resolutions of the

Under this assumption it will be shown that there is an ordered representation of

H onto , Lzlēn , ñ ) relative to T . It will follow from Theorem 10 that U and Ở are

equivalent . Let E and Ể be the resolutions of the

**identity**for T and † respectively .Page 1717

By induction on ( Jil , we can readily show that a formal

Odz = C ( x ) DJ1908 + C1 , 1 , ay ( x ) 20 , II < l , l + lJ2D with suitable coefficients

CJ . J . , holds for every function Cin CO ( 1 . ) . Making use of

...

By induction on ( Jil , we can readily show that a formal

**identity**( 1 ) 2010 ( x )Odz = C ( x ) DJ1908 + C1 , 1 , ay ( x ) 20 , II < l , l + lJ2D with suitable coefficients

CJ . J . , holds for every function Cin CO ( 1 . ) . Making use of

**identities**of the type...

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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

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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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