I seem to recall that (not necessarily closed) two-sided ideals of $B(H)$ are hereditary. Is that true? 


If it is, can anyone post a proof/reference? 



Two-sided ideals of $B(H)$ are hereditary
**Yes**, this is true and it's actually due to Calkin himself. See Theorem 1.6 in



> 
> J. W. Calkin, [Two-Sided Ideals and Congruences in the Ring of Bounded Operators in Hilbert Space](http://www.jstor.org/stable/1968771?seq=1#page_scan_tab_contents), *Annals of Mathematics* Second Series, **42**, No. 4 (Oct., 1941), 839-873.
> 
> 
> 


Calkin proved that ideals of $B(H)$ are in one-to-one correspondence with certain subsets of $c\_0$ that he termed *ideals sets*. Now Theorem 1.6 tells you that the spectral set of a two-sided ideal is an ideal set which, by definition, is hereditary.


It is maybe worthwhile to add that this is not true for ideals of $K(H)$, however I don't have a counter-example off the top of my head.


