Rational matrix pseudodifferential operators

TL;DR

This paper extends the theory of minimal fractional decompositions from scalar rational pseudodifferential operators to matrix operators, providing new insights into their structure and applications in Dirac structures.

Contribution

It introduces the concept of minimal fractional decomposition for matrix pseudodifferential operators and characterizes their properties, advancing the algebraic understanding of these operators.

Findings

Any matrix H in M_n(K(d)) has a minimal fractional decomposition H=AB^(-1).

Any right fractional decomposition is obtained by multiplying A and B on the right by the same non-degenerate element.

Results are applied to the study of maximal isotropicity in Dirac structures.

Abstract

The skewfield K(d) of rational pseudodifferential operators over a differential field K is the skewfield of fractions of the algebra of differential operators K[d]. In our previous paper we showed that any H from K(d) has a minimal fractional decomposition H=AB^(-1), where A,B are elements of K[d], B is non-zero, and any common right divisor of A and B is a non-zero element of 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 K[d]. In the present paper we study the ring M_n(K(d)) of nxn matrices over the skewfield K(d). We show that similarly, any H from M_n(K(d)) has a minimal fractional decomposition H=AB^(-1), where A,B are elements of M_n(K[d]), B is non-degenerate, and any common right divisor of A and B is an invertible element of the ring M_n(K[d]). 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(K [d]). 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.