Dimension of spectral projection subspaces under strong convergence of operators

Question

I have a possibly simple question regarding estimating bounds on spectral projection subspace.

Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$ converging in the strong operator topology to $H_\infty$. I assume that $\dim \big(\text{Im}\chi_{(a-\epsilon,a+\epsilon)}(H_n) \big)=m_n\in \mathbb{N}$ for all $n$. It seems that this should imply that $\dim \Big(\text{Im}\big(\chi_{[a-\epsilon,a+\epsilon]}(H_\infty) \big) \Big)\geq \liminf m_n$. I would also like to say that if $m_n\to \infty$, then $\sigma_{ess}(H_\infty)\cap [a-\epsilon,a+\epsilon]\neq \emptyset$.

Is this result or something of the sort a known result or something that anyone thinks should hold? I want to check whether I can deduce a spectral type statement about the limit in the strong operator topology.

Answer 1

Unfortunately, no. Let $H_n$ be the orthogonal projection onto $\mathbb{C}\cdot e_n$. Then $H_n \to 0$ strongly, but the spectral projection $\chi_{\{1\}}(H_n)$ has dimension $1$ for all $n$.

The second question fails too; let $H_n$ be the spectral projection onto the span of $\{e_n, \ldots, e_{2n}\}$.

Answer 2

This is false. Take $H_n=P_n$ as the projection onto $\ell^2(\{ x: |x|\le n\})$. Then $P_n\to 1$ strongly, $\dim \chi_{\{ 0\} }(P_n)=\infty$, but $\sigma (1) =\{ 1\}$.

What you have in this situation is the opposite semi-continuity: if $H_n\to H$ strongly and $\sigma(H_n)\cap (a-\epsilon,a+\epsilon)=\emptyset$, then also $a\notin\sigma(H)$.