## Linear Operators: Spectral theory |

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Page 898

E is the resolution of the

complex numbers, then E(d)T = TE{6), <j(T3)Qd, where Ts is the restriction of T to

E(d)>0. Proof. The first statement follows from Theorem l(ii). Now for £ £ d it is ...

E is the resolution of the

**identity**for the normal operator T and if 6 is a Borel set ofcomplex numbers, then E(d)T = TE{6), <j(T3)Qd, where Ts is the restriction of T to

E(d)>0. Proof. The first statement follows from Theorem l(ii). Now for £ £ d it is ...

Page 920

Let E and £ be the resolutions of the

Corollary 2.7 it is seen that £ = VEV-1 and hence that F(f) = VF(T)V~l for every

bounded Borel function F. The mapping W = OV of § onto 2n^.i L2(en > A ) is

clearly an ...

Let E and £ be the resolutions of the

**identity**for T and T respectively. FromCorollary 2.7 it is seen that £ = VEV-1 and hence that F(f) = VF(T)V~l for every

bounded Borel function F. The mapping W = OV of § onto 2n^.i L2(en > A ) is

clearly an ...

Page 1717

By induction on \Jt\, we can readily show that a formal

)dr&*+ 2 CjjaAx)?, with suitable coefficients Cj j , holds for every function Cin CTM

(I0). Making use of

By induction on \Jt\, we can readily show that a formal

**identity**(1) P*C(x)dJ* = C{x)dr&*+ 2 CjjaAx)?, with suitable coefficients Cj j , holds for every function Cin CTM

(I0). Making use of

**identities**of the type (1), we may evidently proceed to prove ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 q Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma Proc prove real axis real numbers representation satisfies Section sequence singular solution spectral spectral theory square-integrable subspace Suppose symmetric operator theory topology transform unique unitary vanishes vector zero