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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].

##### MSC:
 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