Additive functions with monotonic norm (Q2707225)

Let \(f\) be an additive function for which \(|f(n)|\) is increasing for \(n\geq n_0\). \textit{K. Kovács} [J. Number Theory 24, 298-304 (1986; Zbl 0597.10052)], generalizing an old result of Erdős, showed that if \(f:\mathbb{Z}\rightarrow \mathbb{R}^d\) then \(f(n)=a\log n\) for some \(a\in \mathbb{R}^d\). She also pointed out that this theorem does not extend to functions \(g:\mathbb{Z}\rightarrow H\), Hilbert space of infinite dimension, by constructing the counterexample defined by \(g_0(p^k)=\pm e_p\sqrt{k\log p}\) where \(e_p\in H\) are orthonormal vectors, for which \(|g_0(n)|=\sqrt{\log n}\). In this paper the author shows the surprising result that any such \(g:\mathbb{Z}\rightarrow H\) is of the form \(g(n)=a\log n+bg_0(n)\) with \(g_0\) as above, \(a\in H\) and \(b\in\mathbb{R}^+\).