## Linear Operators: Spectral theory |

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Page 1079

Show that if 21 , . . . , hn are

a number of times equal to the dimension of the range of E ( 2 ; A ) ) , then the

sequence ...

Show that if 21 , . . . , hn are

**eigenvalues**of A ( each**eigenvalue**à being repeateda number of times equal to the dimension of the range of E ( 2 ; A ) ) , then the

**eigenvalues**of A ( m ) are die dig . . him ij , ia , . . . , im being an arbitrarysequence ...

Page 1383

With boundary conditions A , the

from the equation sin vā = 0 . Consequently , in Case A , the

numbers of the form ( na ) , n 21 ; in Case C , the numbers { ( n + ) } ? , n 20 .

With boundary conditions A , the

**eigenvalues**are consequently to be determinedfrom the equation sin vā = 0 . Consequently , in Case A , the

**eigenvalues**, are thenumbers of the form ( na ) , n 21 ; in Case C , the numbers { ( n + ) } ? , n 20 .

Page 1615

Reference : Rosenfeld , N . S . , The

Differential Operators , Comm . Pure Appl . Math . 13 , 395 - 405 ( 1960 ) . He

proves the following theorem . THEOREM . Let g ( t ) < o be twice continuously

differentiable ...

Reference : Rosenfeld , N . S . , The

**Eigenvalues**of a Class of SingularDifferential Operators , Comm . Pure Appl . Math . 13 , 395 - 405 ( 1960 ) . He

proves the following theorem . THEOREM . Let g ( t ) < o be twice continuously

differentiable ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint operator 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