## Linear Operators: Spectral theory |

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Page 1226

Part ( a ) follows immediately from

immediately from part ( a ) and

) that any symmetric operator with dense domain has a unique minimal closed

symmetric ...

Part ( a ) follows immediately from

**Lemma**5 ( b ) , and part ( b ) followsimmediately from part ( a ) and

**Lemma**5 ( c ) . Q.E.D. It follows from**Lemma**6 ( b) that any symmetric operator with dense domain has a unique minimal closed

symmetric ...

Page 1696

By

has a carrier which is a compact subset Cm of I ... If I is included in a cube D , it

follows from

a ...

By

**Lemma**14 there is a sequence { Fm } of elements of D ( 1 ) , each of whichhas a carrier which is a compact subset Cm of I ... If I is included in a cube D , it

follows from

**Lemmas**13 , 3.13 and 3.12 that there is a unique extension G of F toa ...

Page 1698

Using

by

, so that in case ( a ) we have shown that fopil is the limit in the norm of H ( o ) ( V )

...

Using

**Lemma**2.1 , let y be a function in CO ( V ) with y ( x ) = 1 for x in K. Then ,by

**Lemmas**3.22 and 3.10 , focī = v ( foqi ! ) = lim.m - 2 Whem the norm of HP ( V ), so that in case ( a ) we have shown that fopil is the limit in the norm of H ( o ) ( V )

...

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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 complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure 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