## 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 $5, 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 eba. Since uo is countably additive on 30, uo(eba) = limm uo(een

ba) > k, and so for some m, u0(een) > u0(ee, ba) > k—e. This shows that the set ...

Since Uee, = e, the

**sequence**{eemba, m > 1} is an increasing**sequence**of setswhose union is eba. Since uo is countably additive on 30, uo(eba) = limm uo(een

ba) > k, and so for some m, u0(een) > u0(ee, ba) > k—e. This shows that the set ...

Page 1124

If En, E are in 3% and p(Ea) increases to the limit p(E), then it follows from what

we have already proved that E, is an increasing

E. If Exe is the strong limit of En, then E. s. E and p(Ex) = p(E). Thus, it follows as ...

If En, E are in 3% and p(Ea) increases to the limit p(E), then it follows from what

we have already proved that E, is an increasing

**sequence**of projections and E, sE. If Exe is the strong limit of En, then E. s. E and p(Ex) = p(E). Thus, it follows as ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 other sections not shown

### Other editions - View all

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

### Common terms and phrases

additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero