## Linear Operators: Spectral theory |

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Page 898

If E is the resolution of the

of compler numbers, then E(6)T = TE(3), o(T,) C 3, where T, is the restriction of T

to E(3)S). PRoof. The first statement follows from Theorem 1(ii). Now for § { 5 it is

...

If E is the resolution of the

**identity**for the normal operator T and if 6 is a Borel setof compler numbers, then E(6)T = TE(3), o(T,) C 3, where T, is the restriction of T

to E(3)S). PRoof. The first statement follows from Theorem 1(ii). Now for § { 5 it is

...

Page 920

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

§ onto X.1 L-(3,, pi) relative to T. It will follow from Theorem 10 that U and U are

equivalent. Let E and E be the resolutions of the

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

§ onto X.1 L-(3,, pi) relative to T. It will follow from Theorem 10 that U and U are

equivalent. Let E and E be the resolutions of the

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

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

)0'16", + X CJ.J., a,(2)6', J| <|7|14-J, " ' with suitable coefficients Co.,,, holds for

every function Cin Co(I). Making use of

...

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

**identity**(1) 6/1 C(r)a's = C(a)0'16", + X CJ.J., a,(2)6', J| <|7|14-J, " ' with suitable coefficients Co.,,, holds for

every function Cin Co(I). Making use of

**identities**of the type (1), we may evidently...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 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 |

### Common terms and phrases

additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete 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 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