Stabilizations of the Trotter-Kato theorem and the Chernoff product formula
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Stabilizations of the Trotter-Kato theorem and the Chernoff product formula Sarah McAllister · Frank Neubrander · Armin Reiser · Yu Zhuang
Received: 19 November 2012 / Accepted: 5 February 2013 / Published online: 15 March 2013 © Springer Science+Business Media New York 2013
Abstract This paper concerns versions of the Trotter-Kato Theorem and the Chernoff Product Formula for C0 -semigroups in the absence of stability. Applications to A-stable rational approximations of semigroups are presented. Keywords Trotter-Kato theorem · Chernoff product formula · Operator semigroups · Stability
1 Introduction The Trotter-Kato Theorem and the Chernoff Product Formula play an important role in the mathematical analysis of approximation schemes for strongly continuous (C0 -)semigroups. One of the main ingredients in these fundamental results is the stability of the approximation scheme under consideration. This paper presents variants of these theorems that cover stabilization techniques for intrinsically unstable approximation schemes. First, let us recall one of the statements of the Chernoff
Communicated by Markus Haase. S. McAllister Center for Distance Learning, SUNY Empire State College, Saratoga Springs, NY 12866, USA e-mail: [email protected] F. Neubrander () · A. Reiser Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA e-mail: [email protected] A. Reiser e-mail: [email protected] Y. Zhuang Department of Computer Science, Texas Tech University, Lubbock, TX 79409, USA e-mail: [email protected]
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Product Formula (also known as Lax Equivalence Theorem). Suppose that (A, D(A)) generates a C0 -semigroup T (t) on a Banach space X and let {V (t); t ∈ [0, τ ]} be a family of bounded linear operators with V (0) = I that satisfies the consistency condition V (t)x − x lim = Ax t→0 t for all x in a set D ⊂ D(A) that is dense in X. Then it was shown by Lax and Richtmyer in 1956 [17] (with a stronger consistency condition) and in final form by Chernoff in 1974 [4] that the following statements are equivalent. (i) The family V (t) is stable; that is, there exist ω, M ≥ 0 such that V (t)n ≤ Meωnt for all n ∈ N0 and t ∈ [0, τ ], (ii) limn→∞ V ( nt )n x = T (t)x for all t ≥ 0 and x ∈ X. There are, however, two shortcomings to this result. First, the theorem does not provide any information about the speed of convergence. Second, many consistent schemes become unstable in the nonanalytic case (for exceptions, see [2, Theorem II.1.2] and [3]). Indeed, it was shown by Kato (see [5, p. 224] and [6, p. 77–78]), that the consistent Crank-Nicolson scheme −1 t t VCN (t) := I + A I − A 2 2 is unstable for the shift semigroup T (t)f (x) := f (x + t) on L1 (R) (see also [15, Theorem 3.1.2] for a similar result for the shift semigroup on C0 (R)). We proceed to discuss known results about consistency and stability for rational approximation schemes V (t) := r(tA), where r : C → C is a rational function whose MacLaurin series coincides with the exponential series fo
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