Let $A$ be a square matrix with elements in $\mathbb{R}$ or $\mathbb{C}$, $\rho\left(A\right)$ stands for the spectral radius of $A$, i.e., the maximum absolute eigenvalue of $A$; $A^{*}$ is the conjugate transpose of $A$; an induced matrix norm $\left\Vert *\right\Vert $ besides the usual vector norm properties, has the sub-multiplicative property.

Question: Given a matrix $A$, there exists a matrix norm $\left\Vert *\right\Vert $ such that

$\rho\left(A\right)<1\Rightarrow\left\Vert A\right\Vert <1$ and $\left\Vert A^{*}\right\Vert <1$?

Motivation: In [Horn and Johnson 1985, Matrix Analysis, Lemma 5.6.10] it is shown how to construct a matrix norm such that $\left\Vert A\right\Vert \leq\rho\left(A\right)+\epsilon$ for any given scalar $\epsilon>0$. With this norm it is sufficient to choose $\epsilon$ sufficiently small to guarantee $\left\Vert A\right\Vert <1$. However, it is possible to find examples where any of the above choice of $\epsilon$ leads to $\left\Vert A^{*}\right\Vert >1$. The problem is to find a matrix norm with sub-multiplicative property that guarantees both, $\left\Vert A\right\Vert <1$ and $\left\Vert A^{*}\right\Vert <1.$

Someone know how to prove or a reference with a proof for that?