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Let

$$\mathcal E = \{A \in \mathcal M(n \times n; \mathbb C): \|A\|_2 \le \|A_0\|_2\}$$

where $A_0 $ is some fixed matrix and $\|\cdot\|_2$ denotes the induced $2$-norm. We also have for every $A \in \mathcal E$, $\rho(A)< 1$ where $\rho(\cdot)$ denotes the spectral radius and $\rho(A_0) < 1$. Is it possible to give an upper bound $C$ in terms of $\|A_0\|_2$ such that $\|(I-A)^{-1}\|_2 \le C$ for all $A \in \mathcal E$?

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  • $\begingroup$ Notably, $\mathcal E$ is compact. So, there necessarily exists a stronger upper bound $r < 1$ such that $\rho(A) \leq r$ for all $A \in \mathcal E$. $\endgroup$ – Omnomnomnom Jun 21 '18 at 22:40
  • $\begingroup$ Does $\|A\|_2$ denote the Frobenius norm or the induced 2-norm? $\endgroup$ – Omnomnomnom Jun 21 '18 at 22:41
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    $\begingroup$ @Omnomnomnom: It is induced $2$-norm. $\endgroup$ – user1101010 Jun 21 '18 at 22:42
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The answer is yes. The upper bound I come up with below is $\frac{1}{1 - \|A_0\|_2}$.

It doesn't seem like there's any need to consider the matrix $A_0$ itself. In the below, we will take $M = \|A\|_0 \geq 0$, since no other information about $A_0$ will be used.

Suppose that $M \geq 1$. Then $\mathcal E$ includes the identity matrix, and so we fail to have $\rho(A) < 1$.

Suppose that $M < 1$. We note that $$ \|(I-A)^{-1}\|_2 = \left\|\sum_{k=0}^\infty A^k\right\|_2 \leq \sum_{k=0}^\infty \left\|A^k\right\|_2 \leq \sum_{k=0}^\infty \left\|A\right\|_2^k \leq \sum_{k=0}^\infty M^k = \frac{1}{1 - M} $$

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  • $\begingroup$ This answer simplifies to the simplest answer to the question: Yes! (Sorry, couldn't help myself.) $\endgroup$ – Robert van de Geijn Jun 21 '18 at 22:50
  • $\begingroup$ @ulaff.net I agree that any answer to a well formed yes/no question should include either a yes or a no; I've edited accordingly. $\endgroup$ – Omnomnomnom Jun 21 '18 at 22:53
  • $\begingroup$ @Omnomnomnom: Thanks. I realized I formulated my question in a wrong way. Indeed, the family of interest is for the case $M > 1$ and the family $\mathcal E = \{A \in \mathcal M(n \times n; \mathbb C): \|A\|_2 \le M \text{ and } \rho(A) < 1\}$. Anyway, your answer does addresses the question I asked. $\endgroup$ – user1101010 Jun 21 '18 at 22:59
  • $\begingroup$ @iris2017 you should make a new post with the correctly formulated question, if you haven't done so already $\endgroup$ – Omnomnomnom Jun 21 '18 at 23:02
  • $\begingroup$ @iris2017 also, I'm fairly confident that you'll only get the upper bound you want if you define your family as $$ \mathcal E = \{A : \|A\|_2 \leq M \text{ and } \rho(A) \leq r\} $$ for some fixed $r$ with $0 \leq r < 1$. For this set, it is notable that an upper bound must exist since $\mathcal E$ is compact and the inverse function is continuous. $\endgroup$ – Omnomnomnom Jun 21 '18 at 23:04

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