## Linear Operators: Spectral operators |

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Page 925

16 Let N1, N2, ... be a countable

commuting with each other. Show that there exists a single Hermitian operator T

such that each N, is a Borel function of T. (Hint: Use Theorem 2.1 and Exercise 15

). 17 For ...

16 Let N1, N2, ... be a countable

**sequence**of normal operators in Ś, allcommuting with each other. Show that there exists a single Hermitian operator T

such that each N, is a Borel function of T. (Hint: Use Theorem 2.1 and Exercise 15

). 17 For ...

Page 959

Since Uee, = e, the

whose union is eb, . Since uo is countably additive on 30, uo(eb,) = lim, u0(ee,b,)

> k, and so for some m, uo(een) 2 uo(ee,bn) > k—e. This shows that the set ...

Since Uee, = e, the

**sequence**{ee, b, m > 1} is an increasing**sequence**of setswhose union is eb, . Since uo is countably additive on 30, uo(eb,) = lim, u0(ee,b,)

> k, and so for some m, uo(een) 2 uo(ee,bn) > k—e. This shows that the set ...

Page 1124

That is, q (E) = p(E1) implies E = E1. Similarly, q.(E) < p(E1) implies E s E1. If E, ,

E are in 37 and p(E.) increases to the limit q(E), then it follows from what we have

already proved that E, is an increasing

That is, q (E) = p(E1) implies E = E1. Similarly, q.(E) < p(E1) implies E s E1. If E, ,

E are in 37 and p(E.) increases to the limit q(E), then it follows from what we have

already proved that E, is an increasing

**sequence**of projections and E, s E. If Es, ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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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