Alternate signs Banach-Saks property and real interpolation of operators

In the space of bounded linear operators acting between Banach spaces we define a seminorm vanishing on the subspace of operators having the alternate signs Banach-Saks property. We obtain logarithmically convex-type estimates of the seminorm for operators interpolated by the Lions-Peetre real method. In particular, the estimates show that the alternate signs Banach-Saks property is inherited from a space of an interpolation pair $(A_{0},A_{1})$ to the real interpolation spaces $A_{\theta,p}$ with respect to $(A_{0},A_{1})$ for all $0<\theta<1$ and $1<p<\infty$.