Perhaps it would be reasonable if $||[A_r z_0 \cdots A_r^m z_0] - Y|| < \epsilon$ for some epsilon.

The convention I'm using is that $n \times m$ means $n$ rows and $m$ columns.

I also have defined vector spaces $U, V, W$ of dimensions $n,m,q$ respectively to define an $n \times m \times q$ tensor in $U \otimes V \otimes W$. Although tensors can also be interpreted as linear operators, it seems like this notation may be less acceptable. Would you agree?