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 belongs to an operator ideal, then for any operators and which can be composed with as, then is class as well. Additionally, in order for to be an operator ideal, it must contain the class of all finite-rank Banach space operators.

Formal definition

Let denote the class of continuous linear operators acting between arbitrary Banach spaces. For any subclass of and any two Banach spaces and over the same field, denote by the set of continuous linear operators of the form such that. In this case, we say that is a component of. An operator ideal is a subclass of, containing every identity operator acting on a 1-dimensional Banach space, such that for any two Banach spaces and over the same field, the following two conditions for are satisfied:

Properties and examples

Operator ideals enjoy the following nice properties.
Furthermore, some very well-known classes are norm-closed operator ideals, i.e., operator ideals whose components are always norm-closed. These include but are not limited to the following.