## Linear Operators: Spectral theory |

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Page 1027

5 shows that a is an eigenvalue and hence for some non - zero æ in H we have

Tx = ax , and hence , since T = TE , we have ( ET ) ( Ex ) = 1 Ex . Hence a

to the spectrum of ET . Conversely , suppose that a non - zero scalar 1

5 shows that a is an eigenvalue and hence for some non - zero æ in H we have

Tx = ax , and hence , since T = TE , we have ( ET ) ( Ex ) = 1 Ex . Hence a

**belongs**to the spectrum of ET . Conversely , suppose that a non - zero scalar 1

**belongs**...Page 1116

1 , B

plainly self adjoint and A

= p ( 1 - p / 2 ) - 1 . Thus , by Lemma 9 , T = BA

1 , B

**belongs**to the Hilbert - Schmidt class Cz . If we let Aq ; = ri - P2Pi , then A isplainly self adjoint and A

**belongs**to the class Cr , where r ( 1 - p / 2 ) = p , i . e . , r= p ( 1 - p / 2 ) - 1 . Thus , by Lemma 9 , T = BA

**belongs**to the class Cs , where s ...Page 1684

Then , if every partial derivative of F of order k

every partial derivative of F of order not more than m is continuous in the closure

of Em . PROOF . By Corollary 2 and Hölder ' s inequality , each ( k - m ) th ...

Then , if every partial derivative of F of order k

**belongs**to L ( Em ) , it follows thatevery partial derivative of F of order not more than m is continuous in the closure

of Em . PROOF . By Corollary 2 and Hölder ' s inequality , each ( k - m ) th ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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