A lower and upper bound for the gap metric

A lower bound and an upper bound for the gap metric are presented. These bounds are sharp in the sense that there exist systems reaching the bounds. The lower bound is the H_∞-norm of a rational matrix, and the upper bound is the sum of the lower bound and the norm of a Hankel operator. Only coprime fractional representations over L_∞-matrices are needed to compute the lower bound. Two equivalent forms of the gap metric are presented, and using these it is shown that if the gap of two systems is smaller than 1, then the two directed gaps are the same. An L_∞ -gap metric is introduced as a purely theoretical generalization of the ordinary gap metric. It turns out that the lower bound is the L_∞-gap metric