The Cesaro Operator in ${\mathrm{\ell }}^2$ is Essentially Normal

In this paper we prove that the Cesaro operator $\mathcal{C}$ in ${\mathrm{\ell }}^2$, the Hilbert space of square summable sequences, is essentially normal, i.e. the commutator $[{\mathcal{C}}^{\ast },\mathcal{C}]:={\mathcal{C}}^{\ast }\mathcal{C}-\mathcal{C}{\mathcal{C}}^{\ast }$ is a compact operator on ${\mathrm{\ell }}^2$.