0
$\begingroup$

For Hilbert spaces $H$ and $K$, let $V=B(H,K)(H \neq K)$. A sub space $I$ of $V$ is called an ideal of $V$ if $$IV^*V+VV^*I \subset I$$ and $I$ is called weak ideal of $V$ If $$\text{span}\{IV^*V-VV^*I\} \subset I$$ It is clear that every ideal is weak ideal.

Does there exist a weak ideal of $B(H,K)$ which is not an ideal?

I was trying to construct example from rectangular matrices by having trace in mind but could not get any idea. Any ideas to construct such ideal?

P.S. This question was first posted on MSE but I did not get any answer there.

Improve this question
$\endgroup$
11
  • 1
    $\begingroup$ So $H,K$ are Hilbert spaces? I think you must have made a mistake: if $H,K$ are both separable infinite dimensional (for example) and $V = B(H,K)$ then $V^*V = B(H)$ and $VV^* = B(K)$. Is $I$ means to be a subspace of $B(H,K)$? $\endgroup$
    – Matthew Daws
    Commented Feb 23, 2021 at 19:23
  • 1
    $\begingroup$ @MatthewDaws I think this must all be TRO related, and I suspect the OP has taken definitions pertaining to general TROs and tried to specialize to particular cases. To the OP: perhaps you could give references for the definitions you are presenting here? $\endgroup$
    – Yemon Choi
    Commented Feb 23, 2021 at 19:59
  • $\begingroup$ @YemonChoi: I don’t have references for the ‘weak’ ideal definition, it’s something i am trying to define. As far as ideal is concerned, it’s usually known as TRO ideal. $\endgroup$
    – Math Lover
    Commented Feb 24, 2021 at 3:53
  • $\begingroup$ @MatthewDaws: Sorry you’re right, I means to be sub space of B(H,K). $\endgroup$
    – Math Lover
    Commented Feb 24, 2021 at 3:54
  • 1
    $\begingroup$ Actually, I don't understand the notation, because when you write something like "$V^*V$" I take this to be defined to be a linear span, namely $\operatorname{span}\{ x^*y : x,y\in V \}$. So I don't understand the difference in definition between "ideal" and "weak ideal". $\endgroup$
    – Matthew Daws
    Commented Feb 24, 2021 at 17:10

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .