ON PRODUCTS OF UNBOUNDED COLLECTIONS OF MATRICES

Families A and B of n X n complex matrices with bounded products are s tudied. In particular, it is shown that if A = B-k := {B1B2 ... B-k : B-i epsilon B (i = 1, 2,..., k)} is bounded for some k, then B-m is bo unded for all m greater than or equal to n. The latter result is used to extend the relation lim sup ($) over cap p(k)(B)(1/k)less than or e qual to ($) over cap p(k)(B)(1/k) k-->infinity due to I. Daubechies an d J.C. Lagarias for unbounded families B, where ($) over cap p(k)(B) : = sup{parallel to B-1 B-k parallel to: B-i epsilon B (i = 1,..., k)} a nd k = 1, 2,....