## 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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Page 1967

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

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 first show that e = e * . Since e is the unit , e * = ee * , and so ee * ( ee * ) * e ...Page 2108

We shall say that an

We shall say that an

**element**a € A is scalar ( although Schaefer [ 10 ; p . ... 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 .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

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 algebra ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

28 other sections not shown

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