## Linear Operators: Spectral operators |

### From inside the book

Results 1-3 of 58

Page 1967

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

in X with an

*. Since e is the unit, e* = ee*, and so e = ee* = (ee*)* =e**e* =ee* =e*. Now let x

...

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

in X with an

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

...

Page 2065

Since, for a in Sl^ the function d is continuous on the compact space ®, it follows

that an operator ain^Ij has an

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

%1.

Since, for a in Sl^ the function d is continuous on the compact space ®, it follows

that an operator ain^Ij has an

**inverse**in 91 if d(s) does not vanish onS. Acelebrated theorem of N. Wiener gives more by asserting that the

**inverse**a-1 is in%1.

Page 2069

If the operator A in 9lg has an

" and the determinant S = det(a^) has an ... Since 9l0 contains all

is in 9t0 • It follows from the definition of the determinant of a matrix of operators ...

If the operator A in 9lg has an

**inverse**in 5(§") then, by Corollary 9.6, A ~ x is in 91" and the determinant S = det(a^) has an ... Since 9l0 contains all

**inverses**, S " 1is in 9t0 • It follows from the definition of the determinant of a matrix of operators ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

47 other sections not shown

### Other editions - View all

### Common terms and phrases

adjoint operator algebra of projections Amer analytic arbitrary asymptotic B-space Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator bounded spectral commuting compact complex numbers complex plane constant contains continuous functions converges Corollary countably additive Definition denote differential operator disjoint Doklady Akad domain eigenvalues elements equation equivalent exists finite number Foias follows from Lemma follows from Theorem formal differential operator formula Hence Hilbert space hypothesis identity inequality inverse Lebesgue Math matrix multiplicity Nauk SSSR norm operators in Hilbert perturbation polynomial preceding Proof properties prove quasi-nilpotent restriction Russian satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset Suppose trace class type spectral operator unbounded uniformly bounded unique vector weakly complete zero