## Linear Operators: Spectral theory |

### From inside the book

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Page 1155

orthonormal set in L2(R), which fact was also proved in Theorem 1.6. We leave it

to the reader to show that if R is the compact Abelian group of the real numbers

modulo ...

**Plancherel's theorem**asserts that the set of characters forms a completeorthonormal set in L2(R), which fact was also proved in Theorem 1.6. We leave it

to the reader to show that if R is the compact Abelian group of the real numbers

modulo ...

Page 1158

By the assumption that [ro, m) = 1 for all me R and

- Jr. [ao, mjtf(m)tf(m)u(dm) = so |tf(m)|*u (dm) # 0. On the other hand, by Corollary

3.17 and

By the assumption that [ro, m) = 1 for all me R and

**Plancherel's theorem**we have- Jr. [ao, mjtf(m)tf(m)u(dm) = so |tf(m)|*u (dm) # 0. On the other hand, by Corollary

3.17 and

**Plancherel's theorem**1158 XI. MISCELLANEOUS APPLICATIONS ...Page 1160

measure in R is regular, the set of of functions in L1 on L2(R) which vanish

outside of compact sets in R are dense in L2(R); by

is dense in L2(R) and hence {rf - rgf, ge.}} is dense in L(R). Now let f, g be in 3

and ...

measure in R is regular, the set of of functions in L1 on L2(R) which vanish

outside of compact sets in R are dense in L2(R); by

**Plancherel's theorem**(rffe 3 }is dense in L2(R) and hence {rf - rgf, ge.}} is dense in L(R). Now let f, g be in 3

and ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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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