## 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

and union of two commuting

B ) ...

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 ... Also the ranges of the intersectionand union of two commuting

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

Page 1123

We say that E is a subdiagonalizing

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

a maximal totally ordered set F of orthogonal subdiagonalizing

, a ...

We say that E is a subdiagonalizing

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

a maximal totally ordered set F of orthogonal subdiagonalizing

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

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 ...

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 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 operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently consider 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 multiplicity neighborhood norm obtained partial positive preceding present problem 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 vanishes vector zero