I found this statement (e.g., from here Matrix norm and spectral radius)
For every $A \in \mathbb{R}^{n \times n}$ and $\epsilon > 0$, there exists a matrix norm $\Vert \cdot \Vert$ such that $\Vert A \Vert < \rho(A) + \epsilon$ where $\rho(A)$ is the spectral radius of $A$, i.e., the magnitude of its largest eigenvalue.
I don't need to prove this statement, but I'm having difficulty to see how to prove this and I can't find this in a reference. The important question I want to answer is if all such matrix norms are submultiplicative, i.e., $\Vert A B \Vert \leq \Vert A \Vert \Vert B \Vert$. Ultimately I need to show that $\sum_{t = 0}^{\infty} (A^t)^T Q A^t$ is well defined, where $Q$ is some fixed symmetric positive definite matrix. So it would be very easy if I can argue that all the matrix norms from this statement are submultiplicative.