Wielandt theorem for unital normed algebras

Theorem. (Wielandt (1949)) Let A be a normed unital algebra (with unit e). If x,yA then xy-yxe.


Assume there are x,yA such that xy-yx=e. Then for all n we have

xny-yxn =nxn-10

We prove this by inductionMathworldPlanetmath over n. It holds for n=1 by assumptionPlanetmathPlanetmath. Assume it is valid for n. Then xn0 and

xn+1y-yxn+1 =xn(xy-yx)+(xny-yxn)x

From this identity it follows that

nxn-1 =xny-yxn2xny2xn-1xy

It follows that n2||x||||y|| for all n which is impossible. ∎

Corollary. The identity operator on a Hilbert spaceMathworldPlanetmath cannot be expressed as a commutator of two bounded linear operators in ().

Remark. The above can be understood as a version of the uncertainty principle in one dimension. Let H=L2(). Let q:HH be q(f)(x):=xf(x) with D(q)={fL2():xxf(x)L2()}, the coordinate operator and p:HH,p(f)(x)=-if(x) the momentum operator with D(p):={fL2():fabsolutely  continuous,fL2()}. It follows that

pq-qp =-iidDonD=D(q)D(p)

According to the corollary D=L2() can never be the case.

Title Wielandt theorem for unital normed algebras
Canonical name WielandtTheoremForUnitalNormedAlgebras
Date of creation 2013-03-22 19:01:22
Last modified on 2013-03-22 19:01:22
Owner karstenb (16623)
Last modified by karstenb (16623)
Numerical id 6
Author karstenb (16623)
Entry type Theorem
Classification msc 46H99