## Linear Operators: Spectral theory |

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Page 889

As will be seen in the next section, a

determines a spectral measure which is defined on the Boolean algebra 3 of all

Borel sets in the plane and which satisfies (iv) for every 6 e 3. This spectral

measure ...

As will be seen in the next section, a

**normal operator**T in Hilbert space $)determines a spectral measure which is defined on the Boolean algebra 3 of all

Borel sets in the plane and which satisfies (iv) for every 6 e 3. This spectral

measure ...

Page 898

(T) PRoof. If we put E(0) = 0 when Ó n g(T) is void, then Corollary 4 follows

immediately from Theorem 1 and Corollary IX.3.15. Q.E.D. 5 DEFINITION. The

uniquely defined spectral measure associated, in Corollary 4, with the

(T) PRoof. If we put E(0) = 0 when Ó n g(T) is void, then Corollary 4 follows

immediately from Theorem 1 and Corollary IX.3.15. Q.E.D. 5 DEFINITION. The

uniquely defined spectral measure associated, in Corollary 4, with the

**normal****operator**T ...Page 934

It is trivial to prove that if A and B are commuting self adjoint operators in a Hilbert

space, then AB is self adjoint. It has been seen (cf. Exercise X.8.7) that if A and B

are commuting positive operators, then AB is positive. If A is a

It is trivial to prove that if A and B are commuting self adjoint operators in a Hilbert

space, then AB is self adjoint. It has been seen (cf. Exercise X.8.7) that if A and B

are commuting positive operators, then AB is positive. If A is a

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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