Operator Ideals,Orthonormal Systems and Lacunary Sets

Given an orthonormal system B in some L²(u) we consider the operator ideals II_B and T_B of B-summing and B-type operators and some related ideals. We characterize by certain weak compactness properties when II_B is equal to the operator ideal II₂ of 2-summing operators. In lose that B consists of characters of a compact abelian group we characterize when II_B coincides with the operator ideal II_γ of Gauss-summing operators and when T_B coincides with the operator ideal II_p of type-2 operators. Moreover, we give a necessary and sufficient condition for Fig to contain the operator ideal II_p of p-summing operators (2 < p < ∞) and for T_B to contain the operator ideal Γ_p of p - factorable operators.