## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 1147

COROLLARY : If G is a compact topological group satisfying the second axiom of

countability , and G is not a

G is countable . A complete set of representations of a

COROLLARY : If G is a compact topological group satisfying the second axiom of

countability , and G is not a

**finite**set , then any complete set of representations ofG is countable . A complete set of representations of a

**finite**group is**finite**.Page 1459

Q.E.D. 30 COROLLARY . A formally positive formally symmetric formal differential

operator is

bounded below . Thus the present corollary follows from Corollary 7 and

Definition ...

Q.E.D. 30 COROLLARY . A formally positive formally symmetric formal differential

operator is

**finite**below zero . PROOF . It is obvious from Definition 20 that t isbounded below . Thus the present corollary follows from Corollary 7 and

Definition ...

Page 1460

Then , if t is

t , is

generality that i = 0. By Corollary 24 ( b ) , Corollary XII.4.13 , and Corollary 26 ,

To ( t ) ...

Then , if t is

**finite**below 2 , and the leading coefficient of r + t , never vanishes , t +t , is

**finite**below 2 . Proof . It is clear that we may suppose without loss ofgenerality that i = 0. By Corollary 24 ( b ) , Corollary XII.4.13 , and Corollary 26 ,

To ( t ) ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex 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 matrix measure multiplicity Nauk 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