## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Results 1-3 of 74

Page 1936

... spectral operator if and only if each

T is a spectral operator , then the resolution of the identity for the

X is the corresponding

... spectral operator if and only if each

**restriction**T | E , X is a spectral operator . IfT is a spectral operator , then the resolution of the identity for the

**restriction**T | E ,X is the corresponding

**restriction**of the resolution of the identity for T . PROOF .Page 2094

( X ) is reduced by a closed subspace y 9 X and one of its complements ( that is ,

if T commutes with some projection of X onto Y ) , then the

**Restrictions**and quotients . Theorem 3 . 10 shows that if a spectral operator Te B( X ) is reduced by a closed subspace y 9 X and one of its complements ( that is ,

if T commutes with some projection of X onto Y ) , then the

**restriction**T Y of T to ...Page 2228

If o is a Borel set , and T is a spectral operator with resolution of the identity E ,

then the

resolution of the identity is the

) X is ...

If o is a Borel set , and T is a spectral operator with resolution of the identity E ,

then the

**restriction**T | E ( 0 ) X of T to E ( 0 ) X is a spectral operator whoseresolution of the identity is the

**restriction**of E to E ( 0 ) X . If o is bounded , T | E ( 0) X is ...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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### Common terms and phrases

analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero