Homotopy for the ν-gap

The main result of this paper is to show that a “winding number condition” which relates to two strictly multivariable linear operators, and which is used in the definition of the Vinnicombe (ν-gap) metric, is equivalent to the existence of a multivariable homotopy in the same metric between the two operators. This allows a characterisation of the Vinnicombe metric which is independent of the linearity of the underlying operators, and suggests possible extension of the metric to nonlinear operators.