Uni-asymptotic Linear systems and Jacobi Operators

Abstract

A family left{ U_sright} _{sin S} of bounded linear operators in a normed space X is uni-asymptotic, when all its trajectories left{ U_sxright} _{sin S} with xnot =0 have the same norm-asymptotic behavior (see 1.5); left{ U_sright} _{sin S} is tight, when the operator norm and the minimal modulus of U_s have the same asymptotic behavior (see 1.6). We prove that uni-asymptoticity is equivalent to tightness if dim X<+infty , 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 U_s (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.