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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

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