## 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(6), g(T,) C 5, where T, is the restriction of T

to E(6)\). PRoof. The first statement follows from Theorem I (ii). Now for § { } it is ...

If E is the resolution of the

**identity**for the normal operator T and if Ó is a Borel setof compler numbers, then E(6)T = TE(6), g(T,) C 5, where T, is the restriction of T

to E(6)\). PRoof. The first statement follows from Theorem I (ii). Now for § { } it is ...

Page 920

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

§ onto X., 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., 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

)6/16". H- X CJ.J., a,(r)6', |J| <|J1+|J, with suitable coefficients C,...,,, holds for

every function Cin Co(I). Making use of

...

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

**identity**(1) 3/1 C(r)6/2 = C(r)6/16". H- X CJ.J., a,(r)6', |J| <|J1+|J, with suitable coefficients C,...,,, holds for

every function Cin Co(I). Making use of

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

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete complex numbers continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero