Averaging technique in the periodic decomposition problem

Анотація

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 $\textrm{Ker}[(\textrm{I}-\textrm{T}_1)(\textrm{I}-\textrm{T}_2)]$ can be decomposed as a sum $x_1+x_2$ where $x_k\in \textrm{Ker}(\textrm{I}-\textrm{T}_k), k=1,2$. 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.