Let $\delta_n\in C^0_c(\mathbb{R})$ be a Dirac sequence approximating the Dirac delta "function" $\delta$ with support in $0\in \mathbb{R}$. Then, for each $n$ we have a compact operator $K_n:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ given by multiplication with $\delta_n$. My question is: How changes the spectrum of the compact opearator $K_n$ as $n\rightarrow \infty$ ?

convolutionwith $\delta_n$ (so the spectrum of $K_n$ is the range of the Fourier transform of $\delta_n$). $\endgroup$ – Jochen Glueck Feb 24 at 11:46