## Linear Operators: Spectral theory |

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Page 868

It is seen from Theorem 1 that for each JJ e -o and each as in 3: there is a

3 into the field p of

this ...

It is seen from Theorem 1 that for each JJ e -o and each as in 3: there is a

**complex**number a (JR) such that a H-S) = w(Jo)e-i-J'. This mapping a -> r(S).) of3 into the field p of

**complex**numbers is clearly a homomorphism. Since w()') s athis ...

Page 872

each as in 3 define a (A) = lim P,(A) where {P,} is a sequence of polynomials with

|P,(z)—w -> 0. The number a (A) is clearly independent of the particular ...

**complex**variable that {P,(2)} also converges uniformly on G. For each A in G andeach as in 3 define a (A) = lim P,(A) where {P,} is a sequence of polynomials with

|P,(z)—w -> 0. The number a (A) is clearly independent of the particular ...

Page 887

There we associated with an operator T in a

of sets in the

defined as any subset of the spectrum o(T) which is both open and closed in the ...

There we associated with an operator T in a

**complex**B-space a Boolean algebraof sets in the

**complex**plane which were called spectral sets. A spectral set wasdefined as any subset of the spectrum o(T) which is both open and closed in the ...

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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