## Linear Operators: Spectral theory |

### From inside the book

Results 1-3 of 83

Page 1378

matrix measure { êu ) , i , j = 1 , . . . , k of

= 1 , . . . , k ; Pis = 0 , if i > k or ; > k . PROOF . Suppose that 01 , . . . , 0x is a

determining set for T . Then it is evident from

i , j ...

matrix measure { êu ) , i , j = 1 , . . . , k of

**Theorem**23 is unique , and Pis = Piss i , j= 1 , . . . , k ; Pis = 0 , if i > k or ; > k . PROOF . Suppose that 01 , . . . , 0x is a

determining set for T . Then it is evident from

**Theorem**23 that if we define { Pis } ,i , j ...

Page 1379

{ Ô is } is the matrix measure of

determined for each e C N . Since 1 is the union of a sequence of neighborhoods

of the same type as N , the uniqueness of { u } follows immediately . Q . E . D . 27

...

{ Ô is } is the matrix measure of

**Theorem**23 , the values Pis ( e ) are uniquelydetermined for each e C N . Since 1 is the union of a sequence of neighborhoods

of the same type as N , the uniqueness of { u } follows immediately . Q . E . D . 27

...

Page 1904

15 remarks on , ( 389 – 392 ) Convergence

continuous limits , IV . 6 . 11 ( 268 ) Banach

measurable ...

15 remarks on , ( 389 – 392 ) Convergence

**theorems**, . IV . 15 Alexandroff**theorem**on convergence of measures , IV . 9 . 15 ( 316 ) Arzelà**theorem**oncontinuous limits , IV . 6 . 11 ( 268 ) Banach

**theorem**for operators into space ofmeasurable ...

### What people are saying - Write a review

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

### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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