Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows
\begin{eqnarray*} ||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} \right \} \end{eqnarray*}
where $||.||_2$ denotes the Euclidean norm of the $(NM \times 1)$ vector x, and where $||x||_b$ denotes the block maximum norm of the $(NM \times 1)$ vector x, i.e.
\begin{eqnarray*} ||x||_{2,b} = \mathrm{max}_{1 \leq k \leq N} \left \{ ||x_k||_2 \right \} \end{eqnarray*}
with $x = [x_1^T \, \, x_2^T \ldots \, x_N^T]$ and $x_k$ equal to a $(M \times 1)$ sub-vector of x for all $k$. It is well-known that
$||A||_2 \geq \rho(A)$
and that
$||A||_b \geq \rho(A)$
where $\rho(A)$ denotes the spectral radius of A (i.e. $\mathrm{max} |\lambda_i|$ with $\{\lambda_i\}$ denoting the eigenvalues of the matrix $A$), and where \begin{eqnarray*} ||A||_{2} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_2}{||x||_2} \right \} \end{eqnarray*} and \begin{eqnarray*} ||A||_{b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_b} \right \}. \end{eqnarray*} However, I would like to know if $||A||_{2,b} \geq \rho(A)$. By using the material that there is in the literature, I am only able to prove that $||A||_{2,b} \geq \rho(A)c$ with $c \leq 1$.
