## Linear Operators: Spectral operators |

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Page 1967

a B*-subalgebra X of a B* -algebra 9J has the same unit e as SQ, then an

element in X with an

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

a B*-subalgebra X of a B* -algebra 9J has the same unit e as SQ, then an

element in X with an

**inverse**in 9) has this**inverse**also in X. Proof. We first showthat e = e*. Since e is the unit, e* = ee*, and so e = ee* = (ee*)* = e**e* = ee* = e*.

Page 2065

Since, for a in the function d is continuous on the compact space S, it follows that

an operator a in 9Ij has an

theorem of N. Wiener gives more by asserting that the

Since, for a in the function d is continuous on the compact space S, it follows that

an operator a in 9Ij has an

**inverse**in 21 if d(s) does not vanish on S. A celebratedtheorem of N. Wiener gives more by asserting that the

**inverse**a-1 is in 9T1.Page 2069

Proof, If the operator A in 9Ig has an

is in 91" and the determinant 8 = det(a„) has an

all

Proof, If the operator A in 9Ig has an

**inverse**in B($)p) then, by Corollary 9.6, A " 1is in 91" and the determinant 8 = det(a„) has an

**inverse**in 91. Since 9l0 containsall

**inverses**, 8 " 1 is in 9I0 . It follows from the definition of the determinant of a ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectra Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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