A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some Lp(μ), 1⩽p<∞, where μ is a σ-finite measure. We study the convergence in the Lp(μ)-norm and the almost everywhere convergence of the bilinear operatorsAn(f1,f2)=(12n+1∑i=−nnTif1(x))(12n+1∑i=−nnTif2(x)) for functions f1∈Lp1(μ) and f2∈Lp2(μ), 1⩽p, 1