## Linear Operators: Spectral theory |

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Page 1174

is in none of the sets e. Then (43) | g(s)

the similar well-known equation for scalarvalued functions. This concludes the ...

**Let**h be a scalar-valued function and put g(s) = h(s)a? if s is in e, and g(s) = 0 if sis in none of the sets e. Then (43) | g(s)

**f**(s)is = sh9)**f**(s)'ds; hence (42) follows fromthe similar well-known equation for scalarvalued functions. This concludes the ...

Page 1200

accordance with Definition 5. Corollary 7 shows that [*] f,(T) D fi(T). Let e be a

bounded Borel set of reals so that, by Theorem 3, E(e)) C3 (T) and TE(e) = s. AE(

dź).

**Let f**(T) be this operator and**let f**,(T) be the operator corresponding to finaccordance with Definition 5. Corollary 7 shows that [*] f,(T) D fi(T). Let e be a

bounded Borel set of reals so that, by Theorem 3, E(e)) C3 (T) and TE(e) = s. AE(

dź).

Page 1649

Let I be an open set in E", and

I defined by the equation F(p) = F(j), q e Co(I), is called the compler conjugate of

F. 8 LEMMA. Let I and F be as in the preceding definition, and let r be a formal ...

Let I be an open set in E", and

**let F**be a distribution in I. Then the distribution F inI defined by the equation F(p) = F(j), q e Co(I), is called the compler conjugate of

F. 8 LEMMA. Let I and F be as in the preceding definition, and let r be a formal ...

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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 |

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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 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