## Linear Operators: Spectral operators |

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Page 898

If E is the resolution of the

of compler numbers, then E(0)T = TE(6), g(T,) C 5, where T, is the restriction of T

to E())\}. PRoof. The first statement follows from Theorem I (ii). Now for $ 45 it is ...

If E is the resolution of the

**identity**for the normal operator T and if 0 is a Borel setof compler numbers, then E(0)T = TE(6), g(T,) C 5, where T, is the restriction of T

to E())\}. PRoof. The first statement follows from Theorem I (ii). Now for $ 45 it is ...

Page 920

Let E and E be the resolutions of the

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

bounded Borel function F. The mapping W = ÜV of § onto X.A. L.,(3, pi) is clearly

an ...

Let E and E be the resolutions of the

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

bounded Borel function F. The mapping W = ÜV of § onto X.A. L.,(3, pi) is clearly

an ...

Page 1717

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

6'18", + X. C.J., a j(a)?', |J| <|J1+|Jal with suitable coefficients C,..., holds for every

function Cin CŞ(I). Making use of

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

**identity**(1) 6", C(r)6/2 = C(r)6'18", + X. C.J., a j(a)?', |J| <|J1+|Jal with suitable coefficients C,..., holds for every

function Cin CŞ(I). Making use of

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

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero