## Linear Operators: Spectral theory |

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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 1455

( a ) If T is a closed symmetric operator in Hilbert space which is bounded below

and whose essential spectrum ( T ) does not intersect the interval ( - 0 , 2 ) of the

real axis , we say that T is

( a ) If T is a closed symmetric operator in Hilbert space which is bounded below

and whose essential spectrum ( T ) does not intersect the interval ( - 0 , 2 ) of the

real axis , we say that T is

**finite**below a . ( b ) If į is a formally symmetric formal ...Page 1460

Then , if t is

ry is

generality that a = 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 1 + t never vanishes , r +ry is

**finite**below a . Proof . It is clear that we may suppose without loss ofgenerality that a = 0 . By Corollary 24 ( b ) , Corollary XII . 4 . 13 , and Corollary 26

, To ( t ) ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero