A note on essential spectra and norms of mixed Hankel-Toeplitz operators

Mi~ed sensitivity optimization was considered! by Kwakernaak [9]. Francis et al. [5,4,6] gave various characterizations of the problem, e.g., in terms of the distance from [0 w] to [vM]H °°, where W, M, V, are in H = and M is inner. Jonckheere and Verma [7,12] described the problem in terms of the norm of the Hankel-Toeplitz operator displayed in (1) (below). Implicit methods of minimization, e.g. the e-iteration [6], were introduced by these authors. Apart from the highly implicit nature of the minimization, the theory remains incomplete for irrational plants, for which a method of determining essential spectra has yet to be provided. Here an explicit formula for the essential spectra of such operators will be derived, as well as a method of computing discrete eigenvalues in which the only implicit step involves the evaluation of the zeros of a 'characteristic determinant' function of the real variable h, which is analytic in ?~. The results extend those of Foias et al. [13,2,3] and Flamm [1] for (unmixed) sensitivity minimization. In particular, essential spectra are computed by viewing the operators in question as compact perturbations of multiplication operators, as in [13]. Recently, some results related to the present paper were obtained independently by Juang and Jonckheere [8] but are limited to rational plants.