## Linear Operators: Spectral theory |

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Page 1653

and F in Hot"(I), (cf. Definition 15 (i)). To prove (ii) and the final

lemma, we first note that it is evident for k > 0 from Definition 15 (i). If k < 0 and F

is ...

**Statement**(iii) follows from**statement**(ii) and the fact that |Flori) = |Flø for all k > 0and F in Hot"(I), (cf. Definition 15 (i)). To prove (ii) and the final

**statement**of thelemma, we first note that it is evident for k > 0 from Definition 15 (i). If k < 0 and F

is ...

Page 1756

Hence we find that if |y|<r, f(y) = 0, and

uniqueness of the function V of the theorem is an evident consequence of

existence of the ...

Hence we find that if |y|<r, f(y) = 0, and

**statement**(i) is fully proved. (B) Theuniqueness of the function V of the theorem is an evident consequence of

**statement**(i). Moreover,**statement**(i) enables us to reduce the proof of theexistence of the ...

Page 1771

) follows from

the hypotheses of the present

...

**Statement**(i) follows from the preceding theorem and Theorem 6.23.**Statement**(ii) follows from

**statement**(ii) of the preceding theorem, since a function satisfyingthe hypotheses of the present

**statement**(ii) evidently (cf. Theorem 6.23) satisfies...

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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