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.

Formal definition
Let $$\mathcal{L}$$ denote the class of continuous linear operators acting between arbitrary Banach spaces. For any subclass $$\mathcal{J}$$ of $$\mathcal{L}$$ and any two Banach spaces $$X$$ and $$Y$$ over the same field $$\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$$, denote by $$\mathcal{J}(X,Y)$$ the set of continuous linear operators of the form $$T:X\to Y$$ such that $$T \in \mathcal{J}$$. In this case, we say that $$\mathcal{J}(X,Y)$$ is a component of $$\mathcal{J}$$. An operator ideal is a subclass $$\mathcal{J}$$ of $$\mathcal{L}$$, containing every identity operator acting on a 1-dimensional Banach space, such that for any two Banach spaces $$X$$ and $$Y$$ over the same field $$\mathbb{K}$$, the following two conditions for $$\mathcal{J}(X,Y)$$ are satisfied:
 * (1) If $$S,T\in\mathcal{J}(X,Y)$$ then $$S+T\in\mathcal{J}(X,Y)$$; and
 * (2) if $$W$$ and $$Z$$ are Banach spaces over $$\mathbb{K}$$ with $$A\in\mathcal{L}(W,X)$$ and $$B\in\mathcal{L}(Y,Z)$$, and if $$T\in\mathcal{J}(X,Y)$$, then $$BTA\in\mathcal{J}(W,Z)$$.

Properties and examples
Operator ideals enjoy the following nice properties.


 * Every component $$\mathcal{J}(X,Y)$$ of an operator ideal forms a linear subspace of $$\mathcal{L}(X,Y)$$, although in general this need not be norm-closed.
 * Every operator ideal contains all finite-rank operators. In particular, the finite-rank operators form the smallest operator ideal.
 * For each operator ideal $$\mathcal{J}$$, every component of the form $$\mathcal{J}(X):=\mathcal{J}(X,X)$$ forms an ideal in the algebraic sense.

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.


 * Compact operators
 * Weakly compact operators
 * Finitely strictly singular operators
 * Strictly singular operators
 * Completely continuous operators