ELA, Volume 2, pp. 1-8, April 1997, abstract.
Infinite products and paracontracting matrices
W.-J. Beyn and L. Elsner
In [Linear Algebra Appl., 161:227-263, 1992] the LCP-property of a
finite set Sigma of square complex matrices was introduced and studied.
A set Sigma is an LCP-set if all left infinite products formed from
matrices in Sigma are convergent. It was shown earlier in [Linear Algebra
Appl., 130:65-82, 1990] that a set Sigma paracontracting with respect to
a fixed norm is an LCP-set. Here a converse statement is proved: If Sigma
is an LCP-set with a continuous limit function then there exists a norm
such that all matrices in Sigma are paracontracting with respect to this
norm. In addition the stronger property of l-paracontractivity is
introduced. 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 Sigma consisting of two
projectors the LCP-property is equivalent to l-paracontractivity, even
without requiring continuity.