# Spectrum of Dirac sequences

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 $$n\rightarrow \infty$$ ?

• The multiplication operator is not compact, and its spectrum is the (essential) range of the function you multiply with. – Christian Remling Feb 24 at 9:13
• @ChristianRemling: I think what the OP actually means is that $K_n$ the convolution with $\delta_n$ (so the spectrum of $K_n$ is the range of the Fourier transform of $\delta_n$). – Jochen Glueck Feb 24 at 11:46
• @JochenGlueck: Ok, but then we're still dealing with a multiplication operator after taking FTs, and my comments still apply. – Christian Remling Feb 24 at 17:32
• @ChristianRemling: Fair enough :-). – Jochen Glueck Feb 24 at 19:40