## Linear Operators: Spectral theory |

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Page 865

o,(r) and g(r) are equal. An

where eo is an idempotent with 0 A eo A e clearly has g(t) Co(r). The following

lemma shows that the opposite inclusion holds in case &o has the same unit as 3

.

o,(r) and g(r) are equal. An

**element**r in a B-subalgebra of the form \, = toeowhere eo is an idempotent with 0 A eo A e clearly has g(t) Co(r). The following

lemma shows that the opposite inclusion holds in case &o has the same unit as 3

.

Page 877

Then an

Consequently the spectrum of y as an

as an

Then an

**element**y in §) has an inverse in 3: if and only if it has an inverse in Q).Consequently the spectrum of y as an

**element**of 9) is the same as its spectrumas an

**element**of 3. Proof. If yo-" exists as an**element**of 9) then, since 3 and 9) ...Page 1339

An

equivalence classes of

denoted by L2({u,}). We observe that by Lemma 7, the integrand in the integral ...

An

**element**F of L((u,H) will be said to be a {u,}-null function if |F = 0. The set of allequivalence classes of

**elements**of L({u,}) modulo (u,)-null functions will bedenoted by L2({u,}). We observe that by Lemma 7, the integrand in the integral ...

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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