Operator ideal

In functional analysis, a branch of mathematics, an operator ideal is a special kind of class of continuous linear operators between Banach spaces. If an operator $T$ belongs to an operator ideal ${\mathcal {J}}$, then for any operators $A$ and $B$ which can be composed with $T$ as $BTA$, then $BTA$ is class ${\mathcal {J}}$ as well. Additionally, in order for ${\mathcal {J}}$ to be an operator ideal, it must contain the class of all finite-rank Banach space operators.