Let \((N_ 1,...,N_ n)\) and \((M_ 1,...,M_ n)\) be two operator tuples and \(\Delta (X)=\sum^{n}_{i=1}N_ iXM_ i\), \(\Delta^*(X)=\sum^{n}_{i=1}N^*_ iXM^*_ i\). In this paper we have proved that
(1) Let \((N_ 1,...,N_ n)\) and \((M_ 1,...,M_ n)\) be two nonzero double commuting operator tuples, then for every Hilbert-Schmidt operator X, it holds that \(\| \Delta (X)\|_ 2\geq \| \Delta^*(X)\|_ 2\) if and only if \(N_ 1,...,N_ n\), \(M^*_ 1,...,M^*_ n\) are hyponormal operators.
(2) If \((N_ 1,...,N_ n)\) and \((M_ 1,...,M_ n)\) are commuting normal operator tuples and there is a normal operator N and \(N_ i=f_ i(N)\) \((i=1,...,n)\), where \(f_ i\) \((i=1,...,n)\) are continuous functions, then for every operator X, it holds that \(\| \Delta (X)\|_ 2=\| \Delta^*(X)\|_ 2\).