Uni-asymptotic Linear systems and Jacobi Operators

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 dimX<+∞, 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 detUs (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.