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?
Best Answer
Consider the matrix $A = \pmatrix{0 & 1\cr 0 & 0\cr}$ which has spectral radius $0$. But $A A^*$ has spectral radius $1$, so for any sub-multiplicative norm $$1 \le \|A A^*\| \le \|A\| \|A^*\| \le \max(\|A\|,\|A^*\|)^2$$