Let $T \colon \mathbb{C}^n \to \mathbb{C}^n$ be a linear operator. Let $\{u_k\} \subset \mathbb{C}^n$ and $\{v_k\} \subset \mathbb{C}^n$ be two sequences of vectors. Suppose the spectral radius of $T$ is $\rho(T) = q <1$ but we don't know whether or not the spectral norm $\|T\|_2 = \sup_{\|x\|_2=1} \|Tx\|_2$ is smaller than $1$. I have a recurrence relation defined by \begin{align*} u_{k+1} = T u_k + \beta v_k, \end{align*} where we have the freedom to choose any fixed positive $\beta$. Furthermore, the $2$-norm of $v_k$ is bounded by \begin{align*} \|v_k\|_2 \le \|u_k\|_2 + \|u_{k-1}\|_2, \end{align*}
The problem I am considering is to choose $\beta$ as large as possible but still guarantee $u_k \to 0$.
I have an argument to show there exists some $\beta > 0$ such that $\|u_k\|_2 \to 0$. This argument is somehow involved and the choice of $\beta$ involves the condition number of $S$ where $S^{-1} J S = T$ is the Jordan decomposition.
The argument goes as: We know for any fixed $\varepsilon > 0$ we can define some vector norm $\|\cdot\|_v$ on $\mathbb{C}^n$ such that the induced operator norm of $T$ is $\|T\|_{op,v} = q + \varepsilon$. Then taking $\|\cdot\|_v$ norm on both sides of the recurrence, we get \begin{align*} \|u_{k+1}\|_v \le (q+\varepsilon) \|u_k\|_v + \beta \|v_k\|_v. \end{align*} By norm equivalence, we can bound $\|v_k\|_v \le a \|u_k\|_v + a \|u_{k-1}\|_v$ for some positive constant $a$. Now we have the second order recurrence $\|u_{k+1}\|_v \le (q+ \varepsilon+\beta a) \|u_k\|_v + \beta a \|u_{k-1}\|_v$, which can be solved explicitly. If $\beta$ is sufficiently small, $u_k \to 0$ can be guaranteed. To choose largest $\beta$ with this guarantee, I need the constants $c_1, c_2$ in $c_1\|\cdot\|_2 \le \|\cdot\|_v \le c_2 \|\cdot\|_2$, which is essentially the condition number of $S$.
I am wondering whether there are more direct ways to my problem. I also considered expanding the recurrence \begin{align*} u_{k+1} = T^k u_1 + \beta( v_k + Tv_{k-1} + \dots + T^{k-1} v_1). \end{align*} Then we can apply the fact: for any matrix norm, $\|T^k\| \le c (\rho(T) + \varepsilon)^k$ for every $\varepsilon > 0$ where $c$ is some constant. By taking $2$-norm on both sides, we can avoid what I have done. But I haven't found a way to estimate the sum of $T^{k-j}v_j$ terms. Each sum is defined in a recursive manner with respect to $u_k$ and $u_{k-1}$.
