AbstractGiven an orthonormal systemBin someL2(u) we consider the operator ideals IIBandTBofB‐summing andB‐type operators and some related ideals. We characterize by certain weak compactness properties when IIBis equal to the operator ideal II2of 2‐summing operators. In lose thatBconsists of characters of a compact abelian group we characterize when IIBcoincides with the operator ideal IIγof Gauss‐summing operators and whenTBcoincides with the operator ideal IIpof type‐2 operators. Moreover, we give a necessary and sufficient condition for Fig to contain the operator ideal IIpofp‐summing operators (2