## 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, me .

4%; k(m) = 0, m e C; k(m) = 1, m # 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, me .

4%; k(m) = 0, m e C; k(m) = 1, m # 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 zy, = ovuv, -zy, i.e., xyis 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

* p # 0. From Lemma 12(b) it is seen that g(fo p) Co.(p) and from Lemma 12(c)

and the equation of =7; it

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

**follows**from the above equation that f* p # 0. From Lemma 12(b) it is seen that g(fo p) Co.(p) and from Lemma 12(c)

and the equation of =7; it

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

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 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 Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero