Generalized Young inequality for operator tuples and the Hilbert–Schmidt norm

In this paper, we show that if A=(A1,…,An),B=(B1,…,Bn)∈B(H)n are n-tuples of commuting positive operators, and X∈C2(H), then ‖(∏i=1nAipi)X(∏i=1nBipi)‖2≤‖∑i=1npiAiXBi‖2,where pi>0, i=1,…,n, and ∑i=1npi=1, with equality holding if and only if AiXBi=AjXBj for all i,j∈{1,…,n}. Related inequalities are also discussed.