Submultiplicative function

Summary

A submultiplicative function is a nonnegative function $f$ on a semigroup with $f(xy) \le f(x) f(y)$ for all $x$ and $y$. It is the multiplicative twin of a subadditive function: where $f$ is positive, $\log f$ is subadditive. The operator norm on square matrices is submultiplicative. So is the Frobenius norm, a matrix norm that is not an operator norm. With Fekete's subadditive lemma this gives the existence of limits such as $\lim_n \norm{T^n}^{1/n}$.