## Linear Operators: Spectral theory |

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Page 942

By replacing s by st and u by ut and using the

that Seg ( su - 1 ) q ( ut ) u ( du ) = 10 ( st ) , i . e . , every translate qe of an

eigenfunction o corresponding to a is also an eigenfunction corresponding to 2 .

By replacing s by st and u by ut and using the

**fact**that u ( Et ) = u ( E ) it is seenthat Seg ( su - 1 ) q ( ut ) u ( du ) = 10 ( st ) , i . e . , every translate qe of an

eigenfunction o corresponding to a is also an eigenfunction corresponding to 2 .

Page 1245

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

each complex number a has a unique representation a = reio , where r 2 0 , and

lei = 1 . By analogy with the

...

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

**fact**thateach complex number a has a unique representation a = reio , where r 2 0 , and

lei = 1 . By analogy with the

**fact**that r = ( ão ) , we shall first seek to obtain the self...

Page 1348

are in

the solutions of ro = lo . However , in the range 2 < 0 , at is imaginary , and an

analytic expression like cos att is hard to work with because of the apparent

ambiguity ...

are in

**fact**entire in 2 . In the range a > 0 they form a perfectly suitable basis forthe solutions of ro = lo . However , in the range 2 < 0 , at is imaginary , and an

analytic expression like cos att is hard to work with because of the apparent

ambiguity ...

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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