Quaternionic linear algebra and plurisubharmonic functions of quaternionic variables

Quaternionic linear algebra and plurisubharmonic functions of quaternionic variables. Abstract We remind known and establish new properties of the Dieudonné and Moore determinants of quaternionic matrices. Using these linear algebraic results we develop a basic theory of plurisubharmonic functions of quaternionic variables. The main point of this paper is that in quaternionic algebra and analysis there exist structures which have analogues over the fields of real and complex numbers, but should reflect different phenomena. The algebraic part is discussed in Section 1. There we remind the notions of the Moore and Dieudonné determinants of quaternionic matrices. It turnes out that (under appropriate normalization) the Dieudonné determinant behaves exactly like the absolute value of the usual determinant of real or complex matrices from all points of view (algebraic and analytic). Let us state some of its properties discussed in more details in Subsection 1.2. Let us denote by M n (H) the set of all quaternionic n × n-matrices. The Dieudonné determinant D is defined on this set and takes values in non-negative real number: D : M n (H) −→ R ≥0 1 (see Definition 1.2.2). Then one has the following (known) results (see Theorems 1.2.3 and 1.2.4 below and references given at the beginning of Section 1): Theorem. (i) For any complex n × n-matrix X considered as quaternionic matrix the Dieudonné determinant D(X) is equal to the absolute value of the usual determinant of X. (ii) For any quaternionic matrix X D(X) = D(X t) = D(X *), where X t and X * denote the transposed and quaternionic conjugate matrices respectively. (iii) D(X · Y) = D(X)D(Y).