## 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.