## 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 element in X with 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 ...Page 2065

Since , for a in Aį , the function & is continuous on the compact space S , it follows that an operator a in A , has an

Since , for a in Aį , the function & is continuous on the compact space S , it follows that an operator a in A , has an

**inverse**in A if à ( s ) does not vanish on S. A celebrated theorem of N. Wiener gives more by asserting that the ...Page 2069

If the operator A in 48 has an

If the operator A in 48 has an

**inverse**in B ( HP ) then , by Corollary 9.6 , A - 1 is in AP and the determinant S = det ( ay ) has an**inverse**in A. Since A , contains all**inverses**, 8-1 is in A .. It follows from the definition of the ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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### Common terms and phrases

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 defined Definition denote dense determined differential operator discrete domain elements equation equivalent established exists extension fact finite follows formal formula function 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