## Linear Operators: Spectral theory |

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Page 888

Here we have used the notations A - B and A v B for the intersection and union of

two commuting

and union of two commuting

B ) ...

Here we have used the notations A - B and A v B for the intersection and union of

two commuting

**projections**A and B . We ... Also the ranges of the intersectionand union of two commuting

**projection**operators are given by the equations ( A iB ) ...

Page 1126

Since each

function of T is a strong limit of linear combinations of the

from ( 1 ) that the closure in H ( xm ) of the vectors ( 4 ) is H ( xm ) . Thus , by ...

Since each

**projection**in the spectral resolution of T and hence each continuousfunction of T is a strong limit of linear combinations of the

**projections**Ei , it followsfrom ( 1 ) that the closure in H ( xm ) of the vectors ( 4 ) is H ( xm ) . Thus , by ...

Page 1777

We recall that a

is called an orthogonal

. It has been shown in Lemma 4 that H = MO ( HOM ) where M is an arbitrary ...

We recall that a

**projection**is a linear operator E with E2 = E . A**projection**E in His called an orthogonal

**projection**if the manifolds EH and ( I – EH are orthogonal. It has been shown in Lemma 4 that H = MO ( HOM ) where M is an arbitrary ...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 |

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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 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