## 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 91

Page 1967

If a B * - subalgebra X of a B * - algebra Y has the same unit e as y , then an

show that e = e * . Since e is the unit , e * = ee * , and so e = ee * = ( ee * ) * = e * *

e ...

If a B * - subalgebra X of a B * - algebra Y has the same unit e as y , then an

**element**in X with an inverse in Y has this inverse also in X . PROOF . We firstshow that e = e * . Since e is the unit , e * = ee * , and so e = ee * = ( ee * ) * = e * *

e ...

Page 2108

We shall say that an

used ... The

and the mapping g + g ( a ) = valg ) is an operational calculus for a . We say that a

...

We shall say that an

**element**a e A is scalar ( although Schaefer [ 10 ; p . 143 ]used ... The

**elements**in the range of va are , in a natural sense , “ functions of a ”and the mapping g + g ( a ) = valg ) is an operational calculus for a . We say that a

...

Page 2215

is fundamental in X . Then a bounded operator is in the weakly closed operator

algebra generated by B if and only if it commutes with every

PROOF . It is clear that every

generated ...

is fundamental in X . Then a bounded operator is in the weakly closed operator

algebra generated by B if and only if it commutes with every

**element**of B .PROOF . It is clear that every

**element**in the weakly closed operator algebragenerated ...

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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