hey all,

the basic fact about infinite products of complex numbers is that $\displaystyle \prod_{j=1}^\infty a_j$ is convergent if and only if $\displaystyle \sum_{j=1}^\infty |1-a_j|$ converges. does anyone know if something of this kind holds for matrix-valued case?

I hope for $\displaystyle \prod_{j=1}^\infty A_j$ converges if $\displaystyle \sum_{j=1}^\infty ||A_j-I||<\infty$.

an easier related statement, which looks very true (but I still don't know solution yet) is to prove $\displaystyle \prod_{j=1}^\infty e^{B_j}$ converges if $\displaystyle \sum_{j=1}^\infty ||B_j||<\infty$ (here $\displaystyle B_j$'s are not assumed to commute, of course)

if anyone saw this matter addressed in any book or science paper, I'd appreciate the reference