On One Linear Equation in One Quaternionic Unknown

We study quaternionic linear equations of type λm(x) := m j=1 b j xc j = e with quaternionic constants b j , c j , e and arbitrary positive integer m. For m = 2 the resulting equation is called Sylvester's equation. For this case a complete solution (solution formula, determination of null space) will be given. For the general case we show that the solution can be found by a corresponding matrix equation of a particular simple form. This matrix form is connected with the centralizers of a quaternion and of its isomorphic image in R 4×4. We present a complete determination of these centralizers. However, the mentioned matrix form does not inlude a detection of the singular cases. The determination of singular cases is to some extent possible by applying Banach's fixed point theorem from which we are able to deduce several sufficient conditions for non singular cases. We end the paper with a conjecture on the form of the inverse of a linear mapping and show that interpolation problems and recovery problems have in general no solution. 1. Introduction. Linear mappings in one real or one complex variable are not of much interest from an algebraic point of view. The situation changes if we go to quaternionic linear mappings. In this area non trivial non singular linear mappings exist and we have to cope with the difficulty that it is usually impossible to apriorily distinguish between singular and non singular mappings. Linear quaternionic mappings may be understood as the simplest form of systems of linear equations in quater-nions. Such systems were investigated already by [11, Ore, 1931] with references back to papers of the 19th century. However, all treated equations are of the form a 11 x 1 + a 12 x 2 + · · ·, one coefficient on the left side of the unknowns. In non-commutative algebra also other forms exist, e. g. ax + xd or ax + bxc + xd, and more generally m j=1 b j xc j. A first approach of linear systems including this general type of equation was made by the authors, [8]. However, a thorough investigation of one linear equation in one quaternionic variable is still missing. There is one exception, a paper by [10, R. E. Johnson, 1944] in which an equation of type ax + xd = e was investigated over an algebraic division ring. …