## 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 57

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

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

* = 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 ee * ( ee * ) * e ** e * = ee

* = e * .

Page 2065

Since , for a in Aį , the function â is continuous on the compact space S , it follows

that an operator a in A , has an

celebrated theorem of N. Wiener gives more by asserting that the

in ...

Since , for a in Aį , the function â 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. Acelebrated theorem of N. Wiener gives more by asserting that the

**inverse**a - 1 isin ...

Page 2069

If the operator A in Ap has an

AP and the determinant = det ( av ) has an

of ...

If the operator A in Ap has an

**inverse**in B ( HP ) then , by Corollary 9.6 , A - 1 is inAP and the determinant = det ( av ) has an

**inverse**in A. Since A , contains all**inverses**, 8-1 is in A .. It follows from the definition of the determinant of a matrixof ...

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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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### 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 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