Wold Decomposition in Banach Spaces

We propose a natural analog of the Wold decomposition in the case of a linear noninvertible isometry $V$ in a Banach space $X$. We obtain a criterion for the existence of such a decomposition. In a reflective space, this criterion is reduced to the existence of the linear projection $P\colon X\to V\!X$ with unit norm. Separately, we discuss the problem of the Wold decomposition for the isometry $V_\varphi$ induced by an epimorphism $\varphi$ of a compact set $H$ in the space of continuous functions $C(H)$. We present a detailed study of the mapping $z\to z^m$ of the circle $|z|=1$ with an integer $m\ge2$.