Weakly wandering vectors for unitary actions of commutative groups
Let \(H\) be a Hilbert space, \(G\) a commutative group and \(g \in G \to U_g : H \to H\) a unitary representation or \(G\)-action of \(G\) on \(H\). An non-zero vector \(f \in H\) is weakly wandering with respect to \((U_g)\) if there exists an infinite subset \(S \subset G\) such that for any distinct \(g_1\), \(g_2 \in S\) one has \(\langle U_{g_1} f, U_{g_2} f \rangle = 0\). A non-zero vector \(f\) is called an eigenvector of \((U_g)\) if there exist complex numbers \((\lambda_g)\), \(g \in G\), such that \(U_g f = \lambda_g f\). A unitary representation has continuous spectrum if it has no eigenvectors. The action \((U_g)\) is said to be ``mixing'' if for any \(f_1, f_2 \in H\) \(\lim \langle U_g f_1, f_2 \rangle = 0\) in a precise sense. If \(G = \mathbb{Z}^d\) and \((U_n)\), \(n \in \mathbb{Z}^d\), is a \(\mathbb{Z}^d\)-action on a Hilbert space, it is known, by previous work of the authors and others, that the three following assertions are equivalent: (i) the \(\mathbb{Z}^d\)-action \((U_n)\) has continuous spectrum; (ii) the weakly wandering vectors with respect to \((U_n)\) are dense in \(H\); (iii) the \(\mathbb{Z}^d\)-action \((U_n)\) is ``mixing'' in a precise sense. -- The purpose of this paper is to extend these equivalences, specially (i) and (iii) to more general countable group actions satisfying various mixing conditions.
scientific article; zbMATH DE number 763972