I am working with a product of $n\times n$ matrices $A\_1,\ldots,A\_k$. Under which conditions can I assume that 


$$\|A\_1\cdots A\_k\|\_\infty \leq \|A\_1\cdots \hat{A\_i}\cdots A\_k\|\_\infty \|A\_i\|\_\infty,$$


where $\|\cdot\|\_\infty$ denotes the operator norm, and $\hat{A\_i}$ denotes the omission of $A\_i$.


E.g. if all products of subsets of $\{A\_1,\ldots,A\_k\}$ are normal, then the above inequality should follow from the submultiplicativity of the operator norm and the fact that $AB$ and $BA$ have the same singular values for normal $A$ and $B$. However, this condition seems rather artifical and I am wondering if something more powerful holds. What if all $A\_i$ are normal, self-adjoint, unitary or orthogonal projectors?



Inequality for the operator norm of a product of matrices
For a simple counterexample, suppose $A\_1$, $A\_2$, $A\_3$ are the orthogonal projections
$$ A\_1 = \pmatrix{1 & 0\cr 0 & 0\cr},\ A\_2 = \pmatrix{1/2 & 1/2\cr 1/2 & 1/2\cr},\ A\_3 = \pmatrix{0 & 0\cr 0 & 1\cr}$$
Then $\|A\_1 A\_3\| = 0$ but $\|A\_1 A\_2 A\_3\| > 0$.


That covers all your cases except "unitary". Of course if the $A\_i$ are all unitary, both sides of your inequality are $1$.


