## Linear Operators, Part 2 |

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Page 925

16 Let N 1 , 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).

16 Let N 1 , N2, . . . be a countable

**sequence**of normal operators in Q), 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).

Page 959

Since Ueem = e, the

whose union is eh". Since ,uo is countably additive on Q0, ,uo(eb,,) = limm ;40(ee

,,,b,,) g k, and so for some m, ;t0(ee,,,) g ,a0(ee,,,b,,) > k—s. This shows that the ...

Since Ueem = e, the

**sequence**{ee,,,b,1, m g 1} is an increasing**sequence**of setswhose union is eh". Since ,uo is countably additive on Q0, ,uo(eb,,) = limm ;40(ee

,,,b,,) g k, and so for some m, ;t0(ee,,,) g ,a0(ee,,,b,,) > k—s. This shows that the ...

Page 1124

Similarly, go(E) §(p(E1) implies E g E1. If E", E are in 9' and <p(E,,) increases to

the limit <p(E), then it follows from what we have already proved that E" is an

increasing

then Em ...

Similarly, go(E) §(p(E1) implies E g E1. If E", E are in 9' and <p(E,,) increases to

the limit <p(E), then it follows from what we have already proved that E" is an

increasing

**sequence**of projections and En § E. If Eco is the strong limit of E",then Em ...

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### Contents

SPECTRAL THEORY | 858 |

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 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 f Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero