## Linear Operators: Spectral theory |

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Page 1196

bounded Borel functions into an algebra of normal operators in Hilbert space and

thus the above formula

identity for the self adjoint operator T and let f be a complex Borel function

bounded Borel functions into an algebra of normal operators in Hilbert space and

thus the above formula

**defines**an operational ... Let E be the resolution of theidentity for the self adjoint operator T and let f be a complex Borel function

**defined**...Page 1548

extensions of S and S respectively, and let A,(T) and 2,07) be the numbers

A,(T), n > 1. D11 Let T. be a self adjoint operator in Hilbert space $31, and let To

be a ...

extensions of S and S respectively, and let A,(T) and 2,07) be the numbers

**defined**for the self adjoint operators T and T as in Exercise D2. Show that A, T) >A,(T), n > 1. D11 Let T. be a self adjoint operator in Hilbert space $31, and let To

be a ...

Page 1647

In connection with

identical. Thus, by Lemma 3, a distribution F corresponds to a unique continuous

...

In connection with

**Definition**4, it should be noted that two continuous functions**defined**in I which differ at most on a Lebesgue null set are in fact everywhereidentical. Thus, by Lemma 3, a distribution F corresponds to a unique continuous

...

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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