Communications in Information and Systems

Volume 1 (2001)

Number 4

Homotopy for the $\nu$-gap

Pages: 333 – 380



Thomas S. Brinsmead (RSISE, Australian National University, Canberra, Australia)

Brian D. O. Anderson (RSISE, Australian National University, Canberra, Australia)


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 ($\nu$-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.

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