## Linear Operators: Spectral theory |

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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 No is a Borel function of T. (Hint: Use Theorem 2.1 and Exercise

15).

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

**sequence**of normal operators in S5, allcommuting with each other. Show that there exists a single Hermitian operator T

such that each No is a Borel function of T. (Hint: Use Theorem 2.1 and Exercise

15).

Page 959

Since Uee, = e, the

whose union is eb, . Since uo is countably additive on 30, uo(eb,) = limm u0(een

ba) > k, and so for some m, u0(een) > uo(ee, b, ) > 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,) = limm u0(een

ba) > k, and so for some m, u0(een) > uo(ee, b, ) > 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 q(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 q(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 |

Commutative BAlgebras | 874 |

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Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero