I'm interested in estimates on dimension of spectral projection subspaces of some limit operator. I recently asked a related question in the thread Dimension of spectral projection subspaces under strong convergence of operators. I was shown that the answer is no given just strong convergence, but I was wondering whether additional structure might imply what I want.

Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$. For each $q\in \mathbb{N}$, I can consider $D_q:=\{ -q,...,q \}^2$ and the projection of $\ell^2(\mathbb{Z}^2)$ to functions supported on $D_q$, which I denote by $P_q$. I assume that there exists a sequence of natural $q_n\to \infty$, and there exist matrices $M_n $ such that $ P_{q_n} H_n P_{q_n}=M_n$. If I assume that there exists some $m$ such that,

$$ \dim \Big(\text{Im}\big(\chi_{(a-\epsilon,a+\epsilon)}(M_n) \big) \Big) \geq m \quad \text{for all} \quad n, $$

does it follow that $\dim \Big(\text{Im}\big(\chi_{[a-\epsilon,a+\epsilon]}(H_\infty) \big) \Big)\geq m $, where $H_\infty$ is the strong operator limit of $H_n$?

I hope that if this is not the case, someone will once again have a clever simple argument why this is not true. My question is motivated by a sequence of operators $H_n= \Delta +V_n$, for diagonal operators $V_n$ that agree around bigger and bigger boxes around the origin and $\Delta$ being the discrete Laplacian. These $V_n$ converge strongly to a limit operator $V_\infty$, which gives us $H_\infty=\Delta+V_\infty$. These sort of operators do not allow counter examples as in my previous question in the other thread.

I'm interested in estimates on dimension of spectral projection subspaces of some limit operator. I recently asked a related question in the thread Dimension of spectral projection subspaces under strong convergence of operators. I was shown that the answer is no given just strong convergence, but I was wondering whether additional structure might imply what I want.

Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$. For each $q\in \mathbb{N}$, I can consider $D_q:=\{ -q,...,q \}^2$ and the projection of $\ell^2(\mathbb{Z}^2)$ to functions supported on $D_q$, which I denote by $P_q$. I assume that there exists a sequence of natural $q_n\to \infty$, and there exist matrices $M_n $ such that $ P_{q_{N}} H_n P_{q_{N}}=M_N$ for $n\geq N$(This condition is a later edit). If I assume that there exists some $m$ such that,

$$ \dim \Big(\text{Im}\big(\chi_{(a-\epsilon,a+\epsilon)}(M_n) \big) \Big) \geq m \quad \text{for all} \quad n, $$

does it follow that $\dim \Big(\text{Im}\big(\chi_{[a-\epsilon,a+\epsilon]}(H_\infty) \big) \Big)\geq m $, where $H_\infty$ is the strong operator limit of $H_n$?

I hope that if this is not the case, someone will once again have a clever simple argument why this is not true. My question is motivated by a sequence of operators $H_n= \Delta +V_n$, for diagonal operators $V_n$ that agree around bigger and bigger boxes around the origin and $\Delta$ being the discrete Laplacian. These $V_n$ converge strongly to a limit operator $V_\infty$, which gives us $H_\infty=\Delta+V_\infty$. These sort of operators do not allow counter examples as in my previous question in the other thread.