## Linear Operators, Part 2 |

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Page 907

positive if and only if its spectrum lies on the unit circle, the real axis, or the

Corollary IX.3.15, NN* =N*N = I if and only if '21 = 1 for every spectral point A of N

...

positive if and only if its spectrum lies on the unit circle, the real axis, or the

**non**-**negative**real axis respectively. Paoor. If N is a bounded normal operator then, byCorollary IX.3.15, NN* =N*N = I if and only if '21 = 1 for every spectral point A of N

...

Page 939

By an argument similar to the preceding, there is a

countably additive set function v on Z' with v(G) = 1 and v(Es) = v(E) for E in Z' and

s in G. Let vl be any such function and let /11 be any

measure on ...

By an argument similar to the preceding, there is a

**non**-**negative**regularcountably additive set function v on Z' with v(G) = 1 and v(Es) = v(E) for E in Z' and

s in G. Let vl be any such function and let /11 be any

**non**-**negative**regularmeasure on ...

Page 1088

58 (Mercer) Let the hypotheses of the preceding exercise be satisfied, and

suppose that the operator K is

1). the series converging uniformly. (Hint: Show that K(t, t) = Z,-Ia,-|<p,.(t)|2, and '

hence ...

58 (Mercer) Let the hypotheses of the preceding exercise be satisfied, and

suppose that the operator K is

**non**-**negative**. Show that K (8. 1) = Ea.-<P.(s><P.-(1). the series converging uniformly. (Hint: Show that K(t, t) = Z,-Ia,-|<p,.(t)|2, and '

hence ...

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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