## Linear Operators: Spectral theory |

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Page 984

number e, r(f of)(m) = e for m in a neighborhood N of C. It

preceding lemma that there is a function k in L1(R) n L2(R) with 0 < k(m) < 1, m

evo; k(m) = 0, m e C; k(m) = 1, m of N. Thus k(m)g(m) = 0 for every m in 24 and,

since the ...

number e, r(f of)(m) = e for m in a neighborhood N of C. It

**follows**from thepreceding lemma that there is a function k in L1(R) n L2(R) with 0 < k(m) < 1, m

evo; k(m) = 0, m e C; k(m) = 1, m of N. Thus k(m)g(m) = 0 for every m in 24 and,

since the ...

Page 993

Then it

is independent of V. Q.E.D. 16 THEOREM. If the bounded measurable function

op has its spectral set consisting of the single point m then, for some compler ...

Then it

**follows**from what has just been demonstrated that gy, = zvuv, -zy, i.e., gyis independent of V. Q.E.D. 16 THEOREM. If the bounded measurable function

op has its spectral set consisting of the single point m then, for some compler ...

Page 996

Since f + p is continuous by Lemma 3.1(d) it

f * p + 0. From Lemma 12(b) it is seen that g(for p) Co.(p) and from Lemma 12(c)

and the equation of = of it

Since f + p is continuous by Lemma 3.1(d) it

**follows**from the above equation thatf * p + 0. From Lemma 12(b) it is seen that g(for p) Co.(p) and from Lemma 12(c)

and the equation of = of it

**follows**that o(fo p) contains no interior point of a (p).### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### 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