Dichotomy, discrete Bohl exponents, and spectrum of block weighted shifts (Q1179067)

Suppose \(r\) is a positive integer and \((W_ n)_ n\) a sequence of \(r\times r\)-complex, invertible matrices such that both \((W_ n)\) and \((W_ n^{-1})\) are uniformly bounded. The usual \(C^ r\)- valued \(\ell^ 2\)-space is considered. The block weighted shift operator with weights \((W_ n)\) is the operator having block matrix \((\delta_{i,i-j} W_ j)_{i,j}\). The following system is associated with it \[ x_{n+1}=W_ n x_ n,\qquad n=0,\pm1,\pm2,\dots . \] The discrete Bohl exponents associated with such a system are positive numbers computed in terms of the weights. The notion is analogue to that used in differential equation theory. The spectrum of a block weighted shift is calculated in terms of the corresponding Bohl exponents. Projections satisfying some inequalities involving Bohl exponents are called splitting projections associated to the system. Existence of 1-splitting projections for the system above is shown to be equivalent to the existence of dichotomies.