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 $A \in \mathcal E$?
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 $$ \|(I-A)^{-1}\|_2 = \left\|\sum_{k=0}^\infty A^k\right\|_2 \leq \sum_{k=0}^\infty \left\|A^k\right\|_2 \leq \sum_{k=0}^\infty \left\|A\right\|_2^k \leq \sum_{k=0}^\infty M^k = \frac{1}{1 - M} $$