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On the commutation relation \(AB-BA=I\) for operators on non-classical Hilbert spaces. (English) Zbl 0993.47045
Katsaras, A. K. (ed.) et al., \(p\)-adic functional analysis. Proceedings of the 6th international conference, Ioannina, Greece, July 3-7, 2000. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 222, 177-190 (2001).
The paper is concerned with the validity of the commutation relation \[ AB-BA = I,\tag{1} \] 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 A. Yu. Kochubei [J. Phys. A. Math. Gen. 29, No. 19, 6375-6378 (1996; Zbl 0905.46051)], and, independently, by S. Albeverio and 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.
For the entire collection see [Zbl 0969.00058].

47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
47L10 Algebras of operators on Banach spaces and other topological linear spaces
47L90 Applications of operator algebras to the sciences
46C15 Characterizations of Hilbert spaces