On the S-universal elementary operators

Let B(H)B(H) be the algebra of all bounded linear operators on a Hilbert space H. For n  -tuples of operators A˜=(A1,⋯,An) and B˜=(B1,⋯,Bn) in B(H)B(H), define the elementary operator RA˜,B˜ on B(H)B(H) by RA˜,B˜(X)=∑i=1nAiXBi (X∈B(H)X∈B(H)). We give some upper and lower estimates for the norm of the operator RA˜,B˜ when it is restricted to a given uniform ideal in B(H)B(H). Moreover, we give necessary and sufficient conditions for the norm of the restriction of a sum of two multiplications to a norm ideal, to attain its optimal value.