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

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Page 993

Now let V , be an arbitrary open subset of Ř with compact closure . Then it

from what has just been demonstrated that av , = oyuv , = Qy , i.e. , Qy is

independent of V. Q.E.D. 0 16 THEOREM . If the bounded measurable function o

has its ...

Now let V , be an arbitrary open subset of Ř with compact closure . Then it

**follows**from what has just been demonstrated that av , = oyuv , = Qy , i.e. , Qy is

independent of V. Q.E.D. 0 16 THEOREM . If the bounded measurable function o

has its ...

Page 996

Since f * q is continuous by Lemma 3.1 ( d ) it

that f * 9 +0 . From Lemma 12 ( b ) it is seen that olf * o ) Colo ) and from Lemma

12 ( c ) and the equation of = Tf it

...

Since f * q is continuous by Lemma 3.1 ( d ) it

**follows**from the above equationthat f * 9 +0 . From Lemma 12 ( b ) it is seen that olf * o ) Colo ) and from Lemma

12 ( c ) and the equation of = Tf it

**follows**that o ( f * ) contains no interior point of o...

Page 1124

That is , Q ( E ) = Q ( E ) implies E = E. Similarly , y ( E ) SQ ( E ) implies ESE . If En

, E are in F and q ( En ) increases to the limit ( E ) , then it

have already proved that En is an increasing sequence of projections and En > E

.

That is , Q ( E ) = Q ( E ) implies E = E. Similarly , y ( E ) SQ ( E ) implies ESE . If En

, E are in F and q ( En ) increases to the limit ( E ) , then it

**follows**from what wehave already proved that En is an increasing sequence of projections and En > E

.

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

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### Common terms and phrases

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