# Looking for an example of ‘weak’ ideal

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: Copy of the same question on Mathoverflow can be found here