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This ratio $\|A\|_{2,b}/\rho(A)$ can be arbitrarily small. Consider the $N\times N$ matrix $A$ all of whose entries are equal to $1$, and interpret this as a block matrix with blocks of size $M=1$. Then $\rho(A)=N$, $$ \|Ax\|_b = \max_j |x_1+\ldots + x_N| \le \|x\|_1 , $$ so $\|A\|_{2,b}\le \sup \|x\|_1/\|x\|_2 = \sqrt{N}$.



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