Preprint des Projektes: SFB 343: Diskrete Strukturen in der Mathematik

Theorie und Numerik von Aufgaben der linearen Algebra und diskreter dynamischer Systeme

96-074 Ludwig Elsner, Wolf-Jürgen Beyn.
Connecting Paracontractivity and Convergence of Products

In [2] the LCP-property of a finite set S of square complex matrices was introduced and studied. S is an LCP-set if all left infinite products formed from matrices in S are convergent. It had been shown earlier in [3] that a set S paracontracting with respect to a fixed norm is an LCP-set. Here we prove a converse statement: If S is an LCP-set with a continuous limit function then there exists a norm such that all matrices in S are paracontracting with respect to this norm. In addition we introduce the stronger property of l-paracontractivity. It is shown that common l-paracontractivity of a set of matrices has a simple characterization. It turns out that in the above mentioned converse statement the norm can be chosen such that all matrices are l-paracontracting. It is shown that for S consisting of two projectors the LCP-property is equivalent to l-paracontractivity, even without requiring continuity.