Operator ideals, orthonormal systems and lacunary sets

Given an orthonormal system B in some L-2(mu) we consider the operator idea ls Pi(B) and T-B of B-summing and B-type operators and some related ideals. We characterize by certain weak compactness properties when Pi(B) is equal to the operator ideal Pi(2) of 2 - summing operators. In case that B consi sts of characters of a compact abelian group we characterize when Pi(B) coi ncides with the operator ideal Pi(gamma) of Gauss-summing operators and whe n T-B coincides with the operator ideal T-2 of type-2 operators. Moreover, we give a necessary and sufficient condition for Pi(B) to contain the opera tor ideal Pi(p) of p-summing operators (2 < p < infinity) and for T-B to co ntain the operator ideal Gamma(p) of p-factorable operators.