Noncommutative Analogue of the Moment-to-LRS Implication

Establish whether, for a non-commutative unital ring \mathcal{R}, any matrix A \in \mathrm{Mat}_s(\mathcal{R}) and any \mathcal{R}-linear map \varphi\colon \mathrm{Mat}_s(\mathcal{R}) \to \mathcal{R}, the sequence (\varphi(A^n))_{n\in\mathbb{N}} satisfies a linear recurrence relation in n (i.e., is a linear recurrence sequence of bounded order depending on s), analogous to the commutative case derived from the CayleyHamilton theorem.

Background

Lemma 2.1 shows that for commutative unital rings, the generalized moment sequence (\varphi(A^n)) is an LRS of order s, via the CayleyHamilton theorem.

The authors point out that, although versions of the CayleyHamilton theorem exist for noncommutative rings, they do not directly yield a relation like in the commutative case, leaving the noncommutative analogue uncertain.