## Linear Operators: Spectral theory |

### From inside the book

Results 1-3 of 78

Page 876

Then ( y + Nie ) ( 2 ) y ( 2 ) + Ni = i ( 1 + N ) , and

1 + N ) 2 = \ y + Niel2 = [ ( y + Nie ) ( y + Nie ) * 1 = | ( y + Nie ) ( y - Nie ) | \ y2 +

Nael \ y4I + N2 . Since this inequality must hold for all real N , a contradiction is ...

Then ( y + Nie ) ( 2 ) y ( 2 ) + Ni = i ( 1 + N ) , and

**hence**11+ N = ly + Niel .**Hence**(1 + N ) 2 = \ y + Niel2 = [ ( y + Nie ) ( y + Nie ) * 1 = | ( y + Nie ) ( y - Nie ) | \ y2 +

Nael \ y4I + N2 . Since this inequality must hold for all real N , a contradiction is ...

Page 1027

Ex .

zero scalar belongs to the spectrum of ET . Then , for some non - zero a in EH ,

we have ETx = 2x . Then Tx = hx + y , where y belongs to the subspace ( I - E ) H

...

Ex .

**Hence**a belongs to the spectrum of ET . Conversely , suppose that a non -zero scalar belongs to the spectrum of ET . Then , for some non - zero a in EH ,

we have ETx = 2x . Then Tx = hx + y , where y belongs to the subspace ( I - E ) H

...

Page 1227

and D are clearly linear subspaces of D ( T * ) , it remains to show that the spaces

D ( T ) , Dy , and D are mutually orthogonal , and that their sum is D ( T * ) .

**Hence**T * x = ix , or x e Dt .**Hence**Dt is closed . Similarly , D is closed . Since D4and D are clearly linear subspaces of D ( T * ) , it remains to show that the spaces

D ( T ) , Dy , and D are mutually orthogonal , and that their sum is D ( T * ) .

### What people are saying - Write a review

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

### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

11 other sections not shown

### Other editions - View all

### Common terms and phrases

additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero