Let $\mathcal H$ be a separable Hilbert space, and $\mathbf A: \mathcal H\rightarrow\mathcal H$ a bounded self-adjoint yet non-compact operator. Suppose that I can construct a sequence $(\mathbf A_n)_{n\geq 0}$ of compact self-adjoint operators over $\mathcal H$ that strongly converge (meaning pointwise convergence) to $\mathbf A$.
Since each $\mathbf A_n$ is both compact and self-adjoint, I know that $\mathcal H$ will be the closure of the eigenspaces of each $\mathbf A_n$.
Can I deduce from above that $\mathcal H$ will also be the closure of the eigenspaces of $\mathbf A$?