## Linear Operators: Spectral theory |

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Page 1079

Show that if 27 , . . . , dn are

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

Show that if 27 , . . . , dn 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 Alm ) are 1 , 4 , . . ii , iz , . . . , im being an arbitrary sequence of ...Page 1383

With boundary conditions A , the

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

numbers of the form ( na ) , n = 1 ; in Case C , the numbers { ( n + 3 ) a } ? , n 20 ...

With boundary conditions A , the

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

**eigenvalues**, are thenumbers of the form ( na ) , n = 1 ; in Case C , the numbers { ( n + 3 ) a } ? , 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

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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 remark 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