## 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 B of all

Borel sets in the plane and which satisfies ( iv ) for every de B . This spectral

measure ...

As will be seen in the next section , a

**normal operator**T in Hilbert space Hdetermines a spectral measure which is defined on the Boolean algebra B of all

Borel sets in the plane and which satisfies ( iv ) for every de B . This spectral

measure ...

Page 898

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

this notion of the resolution of the identity with that given in Section 1 we state the

...

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

**normal operator**T is called the resolution of the identity for T . In order to relatethis notion of the resolution of the identity with that given in Section 1 we state the

...

Page 934

7 ) that if A and B are commuting positive operators , then AB is positive . If A is a

commutes with A * . This fact was conjectured by von Neumann , and proofs have

been ...

7 ) that if A and B are commuting positive operators , then AB is positive . If A is a

**normal operator**and if B is an operator which commutes with A , then Bcommutes with A * . This fact was conjectured by von Neumann , and proofs have

been ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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 give 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 remark 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