Rational matrix solutions to the Leech equation: The Ball–Trent approach revisited

Abstract

Abstract. Using spectral factorization techniques, a method is given by which rational matrix solutions to the Leech equation with rational matrix data can be computed explicitly. This method is based on an approach by J.A. Ball and T.T. Trent, and generalizes techniques from recent work of T.T. Trent for the case of polynomial matrix data. IntroductionConsider H ∞ -matrix functions G ∈ H K is positive. Note that (0.1) is equivalent to T G T X = T K and T X ≤ 1. Hence Leech's theorem can be viewed as the analogue of the Douglas factorization lemma [7] within the class of analytic Toeplitz operators. The necessity of (0.2) follows directly from Douglas' factorization lemma and the reformulation of (0.1) in terms of Toeplitz operators. The other implication is more involved. The solution criterion (0.2) can also be formulated directly in terms of the functions G and K, it is equivalent to the mapbeing a positive kernel in the sense of Aronszajn [1], that is, for any finite sequence z 1 , . . . , z n ∈ D the block operator matrix [L(z i , z j )] i,j=1,...,n defines a positive operator on the Hilbert space direct sum of n copies of C m . We note that the actual result by Leech is stated in the general context of Hilbert space operators intertwining shift operators, and in particular holds for operator-valued H ∞ -functions as well. Our interest is primarily in the case where G and K are rational matrix functions.There exists various proofs of Leech's theorem, see [10] and the references therein. In [3] Ball and Trent prove a generalization of Leech's theorem to the polydisc in C d , adapting a technique coined the 'lurking isometry' approach in [2], and give aKey words and phrases. Leech equation, Toeplitz operators, stable rational matrix functions, outer spectral factor. 1 2 SANNE TER HORST description of all X ∈ H ∞ p×q satisfying (0.1). We briefly outline the construction here, specified to the single variable case.The positivity of T G T *. Such a factorization is often referred to as a Kolmogorov decomposition in the literature, cf., [6]. LetΛ • be the analytic operator-valued function on D, with valuesΛ From this identity one derives the existence of a partial isometryThis in turn implies that the function X defined on D byis in H From the point of view of rational matrix functions the above construction has one disadvantage. In general, the Hilbert space H • appearing in (0.5) is infinite dimensional, and in that case it is hard to see when the solution X in (0.7) is rational. In fact, even if both G and K are rational matrix functions,may very well be of infinite rank. More precisely, see Theorem 3.2 below, in the rational matrix case* for all 0 ≤ t ≤ 2π. Overcoming this difficulty is the main theme of the present paper.In the context of the Toeplitz-corona problem, which can be reduced to the special case of (0.1) with q = m and K(z) = I m , z ∈ D, Trent [12] deduced a modification of the above procedure for the special case that G is a row vector (m = 1) polynomial, leading to a rational column vector sol...

State Space Formulas for a Suboptimal Rational Leech Problem II: Parametrization of All Solutions

“…In a few recent publications [21,16,9], a different approach to the Leech problem was presented, also leading to state space formulas for a solution. Although it is not hard to modify this approach to compute a set of rational matrix solutions, it remains unclear at this stage if the method is suitable to compute the set of all solutions, cf., [11].…”

State space formulae for stable rational matrix solutions of a Leech problem

“…We see the above theorem as the state space version of the rational matrix analogue of Theorem 0.1 in [17]. Furthermore, to construct the function F in (1.7)…”

“…we follow the method of proof given in Section 2 of [17], specifying each step in an appropriate state space setting, and using the fact that (1.9) Im…”

“…When G and K are rational, it is known (see [22] or [17]) that condition (1.2) is also necessary and sufficient for the existence of stable rational matrix solutions of (1.1). In the present paper we derive a state space formula for a rational matrix solution whose McMillan degree is at most equal to the McMillan degree of [G K] starting from state space realizations for G and K. Along the way, we obtain a self contained proof of the existence of a rational matrix solution.…”

${\mathscr H}_\infty$ Problem with Nonstrict Inequality and All Solutions: Interpolation Approach

“…We state the following problem. The problem with excluded condition (1.2) is known in the literature as the Leech problem (see [21, p. 107], [17] and [18]) with some recent activity on getting explicit state-space formulas for a solution (see [27], [28], [10], and [11]). Therefore, Problem 1 could be called a two-sided rational Leech problem.…”