Ideal of compact operators

Summary

The compact operators on a Hilbert space $H$ form a set $\mathcal{K}(H)$. It is a norm-closed two-sided ideal of the algebra $\mathcal{B}(H)$ of bounded operators, closed under taking adjoints. It is the norm closure of the finite-rank operators, and it is a C*-algebra in its own right. For a separable infinite-dimensional $H$, it is the only closed two-sided ideal of $\mathcal{B}(H)$ other than $\set{0}$ and $\mathcal{B}(H)$, a theorem of Calkin. The quotient $\mathcal{B}(H)/\mathcal{K}(H)$ is the Calkin algebra, the setting of Fredholm theory.