## Linear Operators: Spectral theory |

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Page 1079

Show that if 12 , . . . , dn are

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

Show that if 12 , . . . , dn are

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

**eigenvalues**of A ( m ) are dig dia . . . dim iz , iz , . . . , im being an arbitrary ...Page 1383

With boundary conditions A , the

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

the numbers of the form ( na ) ? , n 2 1 ; in Case C , the numbers { ( n + 1 ) a } ? , n

...

With boundary conditions A , the

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

**eigenvalues**1 arethe numbers of the form ( na ) ? , n 2 1 ; in Case C , the numbers { ( n + 1 ) a } ? , n

...

Page 1615

Reference : Rosenfeld , N . S . , The

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

proves the following theorem . THEOREM . Let alt ) < 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 alt ) < o be twice continuously

differentiable ...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero