An elementary description of partial indices of rational matrix functions

In his 1972 doctoral dissertation [12], S. Pattanayak formulated an elementary matrix test for the invertibility of a Toeplitz operator with non-singular rational matrix symbol on the unit circle. This criterion is elementary in the sense that the test matrix is explicit in terms of the symbol and requires knowing only the zeros of the determinant of the symbol inside the unit disc. The result of Pattanayak has not received much attention. See, however [4,10]. It develops that an appropriate version of Pattanayak's result formulated for an arbitrary contour leads in a simple manner to an elementary description of the partial (factorization) indices of a non-singular rational matrix function relative to a contour. Formulae for these partial indices were first given by Gohberg, Lerer and Rodman [8,9]. Here we derive an appropriate version of Pattanayak's test and the elementary formulae for the partial indices.