## Linear Operators: Spectral theory |

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Page 907

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

Corollary IX.3.15, NN* = N*N = I if and only if M = 1 for every spectral point 2 of N,

...

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

**non**-**negative**real aris respectively. PRoof. If N is a bounded normal operator then, byCorollary IX.3.15, NN* = N*N = I if and only if M = 1 for every spectral point 2 of N,

...

Page 939

... that there is a uniquely determined countably additive regular measure u on

the family 2 of Borel sets in G for which * = s.s.)a(ds), fe C(G). Since a " is a

positive functional it follows from the Riesz theorem that the measure p is

... that there is a uniquely determined countably additive regular measure u on

the family 2 of Borel sets in G for which * = s.s.)a(ds), fe C(G). Since a " is a

positive functional it follows from the Riesz theorem that the measure p is

**non**-**negative**...Page 1088

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

suppose that the operator K is

series converging uniformly. (Hint: Show that K(t, t) =Xiu, p,(t)*, and hence prove

that ...

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

suppose that the operator K is

**non**-**negative**. Show that K(s,t) = Xu,w,(s)7.0), theseries converging uniformly. (Hint: Show that K(t, t) =Xiu, p,(t)*, and hence prove

that ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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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