## Linear Operators: Spectral theory |

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Page 888

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

two commuting

the equations A i B = AB , A v B = A + B - AB and that the intersection and ...

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

two commuting

**projections**A and B. We recall that these operators are defined bythe equations A i B = AB , A v B = A + B - AB and that the intersection and ...

Page 1123

We say that E is a subdiagonalizing

invariant , i.e. , if ETE = TE . . 3 LEMMA . Any operator T in Hilbert space admits a

maximal totally ordered set F of orthogonal subdiagonalizing

...

We say that E is a subdiagonalizing

**projection**for T if T leaves the range of Einvariant , i.e. , if ETE = TE . . 3 LEMMA . Any operator T in Hilbert space admits a

maximal totally ordered set F of orthogonal subdiagonalizing

**projections**; i.e. , a...

Page 1124

It follows readily that all the

T. We may consequently use the results of Chapter X on the spectral

representation of T to obtain a spectral representation for the maximal family F of

It follows readily that all the

**projections**E in F belong to the spectral resolution ofT. We may consequently use the results of Chapter X on the spectral

representation of T to obtain a spectral representation for the maximal family F of

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure Nauk neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero