## Linear Operators: Spectral theory |

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Page 1175

Then K , is a bounded

each real & o , let Ht , be the

( H 5 , 1 ) ( 5 ) = f ( s ) , 5 > 60 , otherwise . = 0 By Corollary 22 , it follows that ...

Then K , is a bounded

**mapping**of the space L ( L , ( S ) ) into itself . PROOF . Foreach real & o , let Ht , be the

**mapping**in La ( L ( S ) ) defined by the formula ( 47 )( H 5 , 1 ) ( 5 ) = f ( s ) , 5 > 60 , otherwise . = 0 By Corollary 22 , it follows that ...

Page 1671

Then ( i ) F + FoM - 1 is a one - to - one continuous

D ( 12 ) whose inverse is F + FoM ; ( ii ) F→ FoM - 1 is a one - to - one continuous

Then ( i ) F + FoM - 1 is a one - to - one continuous

**mapping**of D ( 11 ) onto all ofD ( 12 ) whose inverse is F + FoM ; ( ii ) F→ FoM - 1 is a one - to - one continuous

**mapping**of A ( k ) ( 11 ) onto all of A ( k ) ( 12 ) ; ( iii ) if all the partial derivatives ...Page 1734

Let U , C1 , be a bounded neighborhood of q chosen so small that BU , CE , and

so that there exists a

the origin such that ( i ) q is one - to - one , is infinitely often differentiable , and q ...

Let U , C1 , be a bounded neighborhood of q chosen so small that BU , CE , and

so that there exists a

**mapping**o of U , onto the unit spherical neighborhood V ofthe origin such that ( i ) q is one - to - one , is infinitely often differentiable , and q ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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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 neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero