Rational matrix pseudodifferential operators

Abstract

The skewfield $\mathcal{K }(\partial )$ of rational pseudodifferential operators over a differential field $\mathcal{K }$ is the skewfield of fractions of the algebra of differential operators $\mathcal{K }[\partial ]$. In our previous paper, we showed that any $H\in \mathcal{K }(\partial )$ has a minimal fractional decomposition $H=AB^{-1}$, where $A,B\in \mathcal{K }[\partial ],\,B\ne 0$, and any common right divisor of $A$ and $B$ is a non-zero element of $\mathcal{K }$. Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-zero element of $\mathcal{K }[\partial ]$. In the present paper, we study the ring $M_n(\mathcal{K }(\partial ))$ of $n\times n$ matrices over the skewfield $\mathcal{K }(\partial )$. We show that similarly, any $H\in M_n(\mathcal{K }(\partial ))$ has a minimal fractional decomposition $H=AB^{-1}$, where $A,B\in M_n(\mathcal{K }[\partial ]),\,B$ is non-degenerate, and any common right divisor of $A$ and $B$ is an invertible element of the ring $M_n(\mathcal{K }[\partial ])$. Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-degenerate element of $M_n(\mathcal{K } [\partial ])$. We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures.