Uni-asymptotic Linear systems and Jacobi Operators

Abstract

A family {Us} s∈S of bounded linear operators in a normed space X is uni-asymptotic, when all its trajectories {Usx} s∈S with x = 0 have the same norm-asymptotic behavior (see 1.5); {Us} s∈S is tight, when the operator norm and the minimal modulus of Us have the same asymptotic behavior (see 1.6). We prove that uni-asymptoticity is equivalent to tightness if dim X < +∞, and that the finite dimension is essential. Some other conditions equivalent to uni-asymptoticity are provided, including asymptotic formulae for the operator norm and for the trajectories, expressed in terms of determinants det Us (see Theorem 1.7). We find a connection of these abstract results with some results and notions from spectral theory of Jacobi operators, e.g., with the H-class property for transfer matrix sequence.Mathematics Subject Classification. 47D06, 47B36, 81Q10.1 In this paper we identify indexed family {fs} s∈S with the function on S given by: S s −→ fs. 23 Page 2 of 15 M. Moszyński IEOTAbove S represents the set of "admissible moments of time", and X-the set of "admissible states" for the process. Hence, in this paper, for fixed x ∈ X, we use the name trajectory (or orbit), when we consider the family {U s x} s∈S of elements of X, obtained by a family {U s } s∈S of operators. Let us stress, however, that the actual role of the mathematical objects X, S, {U s } s∈S and {U s x} s∈S can be essentially different than their typical role as above-that is, as in mathematical descriptions of some real processes, e.g, in physics etc. As we shall see in Sect. 2 (see 2.28), those objects can be also useful for mathematical description of some "purely mathematical" processes for purely mathematical goals.In this paper we consider only the linear case, i.e., X is a Banach space here and all the operators U s are linear (and we shall mainly assume finite dimension of X). However, some notions introduced here have also sense in non-linear case, and they could be worth future research.Section 1 is devoted to some abstract studies of so-called uni-asymptotic families of operators, i.e., such families {U s } s∈S that all its non-zero trajectories {U s x} s∈S have "the same norm-asymptotic behavior". This precisely means, that for any non-zero initial conditions x, y ∈ X we have3). Although the word "asymptotic" may not fit well to the above definition for the general S, it makes sense for some ordered S-s, e.g., for S = N n0 (see 0.1). The name "uni-asymptotic" is used also for linear spaces consisting of some functions f = {f (s)} s∈S from S into X, with a natural analogic meaning. This section contains the main results of the paper-Theorems 1.5 and 1.7 on some equivalent conditions for uni-asymptoticity in finite dimension. One of the most important observations is that the uni-asymptoticity is equivalent to tightness, where {U s } s∈S is tight, when the operator norm and the minimal modulus of U s have the same asymptotic behavior (see 1.6). Examples showing the necessity of the dim X < ∞ assumption are also provided. They show as well, th...