## Linear Operators: Spectral theory |

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Page 888

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

two commuting

union of two commuting

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

two commuting

**projections**A and B. We ... Also the ranges of the intersection andunion of two commuting

**projection**operators are given by the equations (A a ...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

marimal totally ordered set 3% of orthogonal subdiagonalizing

totally ...

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

marimal totally ordered set 3% of orthogonal subdiagonalizing

**projections**; i.e., atotally ...

Page 1126

Since each

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

from (1) that the closure in §(a,n) of the vectors (4) is $)(a,n). Thus, by taking m ...

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**E, it followsfrom (1) that the closure in §(a,n) of the vectors (4) is $)(a,n). Thus, by taking m ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 other sections not shown

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additive Akad algebra Amer analytic applied 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 determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math measure multiplicity 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