## Linear Operators: Spectral theory |

### From inside the book

Results 1-3 of 84

Page 1187

The inverse of a

only if its domain is

which maps ( x , y ) into [ y , x ] then I ( T - 1 ) = A I ( T ) which shows that T is ...

The inverse of a

**closed**operator is**closed**. A bounded operator is**closed**if andonly if its domain is

**closed**. PROOF . If A , is the isometric automorphism in H ^ Hwhich maps ( x , y ) into [ y , x ] then I ( T - 1 ) = A I ( T ) which shows that T is ...

Page 1226

Q . E . D . It follows from Lemma 6 ( b ) that any symmetric operator with dense

domain has a unique minimal

make the following definition . 7 DEFINITION . The minimal

Q . E . D . It follows from Lemma 6 ( b ) that any symmetric operator with dense

domain has a unique minimal

**closed**symmetric extension . This fact leads us tomake the following definition . 7 DEFINITION . The minimal

**closed**symmetric ...Page 1436

2 , every finite dimensional subspace of a B - space is

HahnBanach theorem ( II . 3 . 13 ) there exists a set xm , . . . , of continuous linear

functionals on the B - space such that x * ( 93 ) = 0 for 0 Si i sk , X ( : ) = 1 . Let D =

{ x ...

2 , every finite dimensional subspace of a B - space is

**closed**. Thus , by theHahnBanach theorem ( II . 3 . 13 ) there exists a set xm , . . . , of continuous linear

functionals on the B - space such that x * ( 93 ) = 0 for 0 Si i sk , X ( : ) = 1 . Let D =

{ x ...

### What people are saying - Write a review

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

### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

36 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 satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero