# Dimension of spectral projection subspaces under strong convergence of operators

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.

## 2 Answers

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 $$\{e_n, \ldots, e_{2n}\}$$.

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 $$a\notin\sigma(H)$$.