## Linear Operators: Spectral theory |

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Page 925

16 Let Nlt N2, . . . be a countable

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

each Nk is a Borel function of T. (Hint: Use Theorem 2.1 and Exercise 15). 17 For

...

16 Let Nlt N2, . . . be a countable

**sequence**of normal operators in all commutingwith each other. Show that there exists a single Hermitian operator T such that

each Nk is a Borel function of T. (Hint: Use Theorem 2.1 and Exercise 15). 17 For

...

Page 959

Since \Jeem = e, the

whose union is ebn. Since fi0 is countably additive on 3S0, fi0(ebn) = limm fi0(

eembn) ^ A;, and so for some m, fi0(eem) ^ /u0(eembn) > k—e. This shows that

the ...

Since \Jeem = e, the

**sequence**{eembn, m ^ 1} is an increasing**sequence**of setswhose union is ebn. Since fi0 is countably additive on 3S0, fi0(ebn) = limm fi0(

eembn) ^ A;, and so for some m, fi0(eem) ^ /u0(eembn) > k—e. This shows that

the ...

Page 1124

Hence, if we choose a countable set Q ^F such that {^(E^} is dense in the range

of the function <p, then each E in F is the limit either of an increasing or of a

decreasing

...

Hence, if we choose a countable set Q ^F such that {^(E^} is dense in the range

of the function <p, then each E in F is the limit either of an increasing or of a

decreasing

**sequence**of projections in F . We shall show below that there exists a...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive 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