Stability for Lie–Trotter products for some operator matrix semigroups

Abstract

Coupled systems of linear differential equations in Banach spaces can be often handled by the theory of C0-semigroups of operator matrices. We study the stability of Lie-Trotter products of such matrix semigroups, and present three classes of examples (abstract delay equations, abstract inhomogeneous equations, abstract dynamic boundary value problems) and some open problems. This survey is based on the papers [1], [2] and [5], to which we refer the interested reader for more details and extensive bibliographical information.The Lie-Trotter product formula provides the motivation and the fundamental background for operator splitting schemes in numerical analysis. Given linear operators A, B on a Banach space E one is interested in the solution of the Cauchy probleṁ u(t) = (A + B)u(t) with fixed initial value u(0) = u 0 . The general theory of C 0 -semigroups yields that the problem is well-posed if and only if A + B is the generator of a C 0 -semigroup (U (t)) t≥0 , and that in this case the unique solution is given by u(t) = U (t)u 0 (we refer to standard monographs on semigroup theory, such as [7]). Now suppose that A, B generate the C 0 -semigroups (S(t)) t≥0 , (T (t)) t≥0 , respectively. If (λ − A − B) has dense range and domain D(A + B), then one has the following equivalence, stating the Lie-Trotter, or sequential splitting, formula: Stability:There are M, ω ≥ 0 with S t n T t n n ≤ M e ωt for all t ≥ 0, n ∈ N.The expressions S t n T t n n are called Lie-Trotter products and, as we see, their convergence is equivalent to their stability.This is an instance of the Lax equivalence theorem, and because of this equivalence we shall concentrate on stability issues. Every C 0 -semigroup (T (t)) t≥0 obeys an exponential estimate of the kind T (t) ≤ M e ωt for some M, ω ≥ 0, and is called quasicontractive if M can be take to be 1. If A, B both generate quasicontractive semigroups, then the stability of the corresponding Lie-Trotter products is a simple consequence of the submultiplicativity of the operator norm. In some cases the space E can be renormed such that both semigroups become quasicontractive, in which case the stability is again immediate. This is certainly so if B ∈ L (E), i.e., if B is bounded, since we can renorm E such that the semigroup (S(t)) t≥0 generated by A becomes quasicontractive, but a semigroup with bounded generator is quasicontractive.Proposition 1 (see [1, Prop. 2.4]) Let A be a generator and let B be bounded, then the corresponding Lie-Trotter products are stable.The importance of splitting procedures becomes inevitable if one considers Cauchy problems (on products E ×F of Banach spaces) given by operator matriceṡIn general, A can be represented only formally as an operator matrix: Its domain is usually nondiagonal (i.e., contains a certain coupling of the two coordinates). That is why, even if C = 0 and D = 0 the semigroup generated by A may not be of diagonal but rather of upper triangular form. Here is a characterization of such matrix semigroups:Proposition 2 (see [1, Prop. 2.1]) Le...