Wold decomposition in Banach spaces (Q941917)

A Wold decomposition generated by a linear isometry \(V\) on a Hilbert space \(X\) is an infinite orthogonal direct sum \(X=M\oplus K\) of \(X\) with \(K=\bigoplus_{0}^\infty V^{n}R\), where \(R\) is the orthogonal complement of the image \(VX\), the subspaces \(M,K\) are invariant with respect to \(V\), the restriction of \(V\) to \(M\) is unitary and the restriction of \(V\) to \(K\) does not have eigenvectors. Moreover, \(V\) is not unitary on any invariant subspace \(L\) of \(K\), or, in other words, the restriction of \(V\) to \(K\) is totally non-unitary. It is well known that in the Hilbert case any linear isometry admits a Wold decomposition (see, e.g., the book [\textit{B. Szőkefalvi-Nagy} and \textit{C. Foiaş}, Analyse harmonique des opérateurs de l'espace de Hilbert. Budapest: Akadémiaí Kiadó; Paris: Masson et Cie (1967; Zbl 0157.43201)]). The aim of the present paper is to extend Wold decompositions to Banach spaces by replacing the inner-product orthogonality by Birkhoff orthogonality: if \(X_1,X_2\) are subspaces of a Banach space \(X\), then one writes \(X_1\perp X_2\) provided that for all \(x_1\in X_1\), \(x_2\in X_2\), \(\|x_1\|\leq \|x_1+x_2\|\). A norm-one linear projection \(P: X\to X_1\) is called an orthogonal linear projection. For a linear isometry \(V:X\to X\), one supposes that there exists an orthogonal linear projection \(P:X\to VX\). Putting \(R=(I-P)X\), it follows that \(V^nR\perp R\) for every \(n\geq 1.\) Starting from these operators, the author defines the analog of the Wold decomposition of the Banach space \(X:X=M\oplus K\), \(K=\bigoplus_0^\infty V^nR\) and gives in Theorem 2.1 some sufficient conditions for the existence of a Wold decomposition. As a particular case, one obtains that any linear isometry on \(\ell^p\), \( 1<p<\infty\), generates a Wold decomposition. The results are applied to the case of isometries \((Vx)(\xi)=x(\varphi \xi)\), \(\xi\in H,\) of the space \(C(H)\) of continuous functions on the compact space \(H\), where \(\varphi: H\to H\) is a homeomorphism, with a detailed examination of the case of the mapping \(z\mapsto z^m\), \(m\in \mathbb N\), \(m\geq2,\) on the unit circle \(|z|=1.\)