Averaging technique in the periodic decomposition problem

Let $T_1$, $T_2$ be a pair of commuting isometries in a Banach space $X$. Generalizing results of M. Laczkovich and Sz. Revesz we prove that in many cases element $x$ of $\mathrm{Ker}[(I-T_1)(I-T_2)]$ can be decomposed as a sum $x_1+x_2$ where $x_k\in\mathrm{Ker}(I-T_k)$, $k=1,2$. Moreover, using an averaging technique we prove the existence of linear operators perfoming such a representation. The results are applicable for decomposition of functions into a sum of periodic ones.