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: X → VX with unit norm. Separately, we discuss the problem of the Wold decomposition for the isometry Vφ induced by an epimorphism φ of a compact set H in the space of continuous functions C(H). We present a detailed study of the mapping z → zm of the circle |z| = 1 with an integer m ≥ 2.