4
$\begingroup$

Let $A$ and $B$ be positive commuting bounded operators on a Hilbert space. It can be shown by functional calculus that $AB=A^{1/2}BA^{1/2},$ so that $AB$ is again positive. If $A$ and $B$ are not bounded, it is known to be false. (Can be easily proved by showing the existence of a certain "bad" $*$-representation of $\mathbb{C}[x,y]$.) It would be good to have an explicit example of such operators (e.g. differential operators on the Schwarz space). My question is:

give an example of linear operators $A,B$ on a (infinite-dimensional, complex) unitary space $V$ such that: 1) $\langle A\varphi,\varphi\rangle\geq 0,\ \langle B\varphi,\varphi\rangle\geq 0, \forall\varphi\in V;$
2) $AB\varphi=BA\varphi,\ \forall\varphi\in V;$ 3) $\langle AB\psi,\psi\rangle< 0$ for some $\psi\in V.$

Unfortunately, I even have no link for the existence of such operators.

$\endgroup$
5
  • 1
    $\begingroup$ I have to confess that I do not completely understand your question. An oprator $A:V\to V$, which is positive, has to be bounded by Hellinger-Toeplitz. If $A$ and $B$ are not everywhere defined, unbounded commuting selfadjoint operators, then the product is still positive, but not necessarily selfadjoint. Is this your question? $\endgroup$ Feb 25, 2011 at 19:55
  • $\begingroup$ Ups, I made a mistake, of course the product will be selfadjoint, even in the unbounded case. Sorry, sorry. So now I am really lost... Maybe I am a bit slow. $\endgroup$ Feb 25, 2011 at 20:08
  • $\begingroup$ Does "unitary space" you mean "inner-product space" (i.e. differs from a Hilbert space by being not complete)? $\endgroup$ Feb 25, 2011 at 20:31
  • $\begingroup$ yes, I mean inner-product space $\endgroup$
    – yurius
    Feb 26, 2011 at 20:57
  • $\begingroup$ Of course, if $V$ is a Hilbert space, then Hellinger-Toeplitz applies $\endgroup$
    – yurius
    Feb 26, 2011 at 20:58

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.