On the commutation relation \(AB-BA=I\) for operators on non-classical Hilbert spaces (Q2751733)

The paper is concerned with the validity of the commutation relation NEWLINE\[NEWLINEAB-BA = I,\tag{1}NEWLINE\]NEWLINE for operators on non-archimedean (n.a.) Banach spaces. It is known that the equality \(xy-yx = e\) is impossible in any Banach algebra with unit over \(\mathbb R\) or \(\mathbb C\). The situation is different in the n.a. case. The first example of this kind was given by \textit{A. Yu. Kochubei} [J. Phys. A. Math. Gen. 29, No. 19, 6375-6378 (1996; Zbl 0905.46051)], and, independently, by \textit{S. Albeverio} and \textit{A. Khrennikov} [J. Phys., A. Math. Gen. 29, No. 17, 5515-5527 (1996; Zbl 0903.46073)], namely, there exists two continuous linear operators on \(c_0\) satisfying (1). The author extend this result characterizing the Banach spaces for which (1) holds. They work with Banach spaces over complete n.a. valued field \(\mathbb K\) having an orthogonal base and for which the non-zero norm values lie in the Dedekind completion of the value group of \(\mathbb K\). As corollary, they obtain similar characterizations for norm Hilbert spaces and for certain Banach spaces whose norms take values in more general sets.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].