## Linear Operators: Spectral theory |

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Page 1220

In particular it is seen, by placing f = a1, that E(e)al = (E(e)a), = 0, which

contradicts the

suppose that the theorem is proved for n < p where p is a positive integer at most

equal to the ...

In particular it is seen, by placing f = a1, that E(e)al = (E(e)a), = 0, which

contradicts the

**fact**that u(e) # 0 and proves the theorem in case m = 1. Nowsuppose that the theorem is proved for n < p where p is a positive integer at most

equal to the ...

Page 1245

This result may be regarded as a far-reaching generalization of the

complex number 2 has a unique representation 2 = re'", where r > 0, and le" = 1.

By analogy with the

This result may be regarded as a far-reaching generalization of the

**fact**that eachcomplex number 2 has a unique representation 2 = re'", where r > 0, and le" = 1.

By analogy with the

**fact**that r = (5x)}, we shall first seek to obtain the self adjoint ...Page 1348

are in

solutions of to - Ao. However, in the range 2 < 0, 2} is imaginary, and an analytic

expression like cos A*t is hard to work with because of the apparent ambiguity in

...

are in

**fact**entire in A. In the range A - 0 they form a perfectly suitable basis for thesolutions of to - Ao. However, in the range 2 < 0, 2} is imaginary, and an analytic

expression like cos A*t is hard to work with because of the apparent ambiguity in

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 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 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