Uni-asymptotic Linear systems and Jacobi Operators

摘要

A family {U-s}(s subset of S) of bounded linear operators in a normed space X is uni-asymptotic, when all its trajectories {UsX}(s is an element of S) with x not equal 0 have the same norm-asymptotic behavior (see 1.5); {U-s}(s is an element of 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 < +infinity, 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.