Recover approximate monotonicity of induced norms

MathOverflow Asked by ippiki-ookami on November 3, 2020

Let $$A$$ some square matrix with real entries.
Take any norm $$|cdot|$$ consistent with a vector norm.

Gelfand’s formula tells us that $$rho(A) = lim_{n rightarrow infty} |A^n|^{1/n}$$.

Moreover, from [1], for a sequence of $$(n_i)_{i in mathbb{N}}$$ such that $$n_i$$ is divisible by $$n_{i-1}$$, we also know that the sequence $$|A^{n_i}|^{1/n_i}$$ is monotone decreasing and converges towards $$rho(A)$$. I am interested in what happens when this divisibility property is not verified.

1. If the matrix has non-negative entries, it seems the general property holds: For integers $$n$$ and $$m$$ such that $$m > n$$, it is the case that $$|A^m|^{1/m} leq |A^n|^{1/n}$$.

2. If the matrix can have positive and negative entries, this more general observation does not seem to hold. I am trying to understand why it fails, how worse can the inequality become, and if it is possible to recover an inequality up to some function of $$A$$: $$|A^m|^{1/m} leq f(A)cdot|A^n|^{1/n}$$.

Any references to 1., or pointers for understanding 2. would be much appreciated.

[1] Yamamoto, Tetsuro. "On the extreme values of the roots of matrices." Journal of the Mathematical Society of Japan 19.2 (1967): 173-178.

This is not a complete answer: If you allow positive and negative entries then this monotonicity will not hold in general. Consider $$A = left[begin{matrix} 0 & 1 & -1 & 0 & 0 \ 0& 0 & 1&1 & 1 \ 0 & 0 & 0 & 1 & 1 \ 0 & 0 & 0 & 0 & -1 \ 0& 0&0 &0&0 end{matrix}right]$$ then $$A^2 = left[begin{matrix} 0 & 0 & 1 & 0 & 0 \ 0& 0 & 0&1 & 0 \ 0 & 0 & 0 & 0 & -1 \ 0 & 0 & 0 & 0 & 0 \ 0& 0&0 &0&0 end{matrix}right] textrm{and} A^3 = left[begin{matrix} 0 & 0 & 0 & 1 & 1 \ 0& 0 & 0&0 & -1 \ 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 \ 0& 0&0 &0&0 end{matrix}right].$$ Thus, $$|A^3|^{1/3} > 1= |A^2|^{1/2}$$.