Operator ideal

Formal definition

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

(1) If ${\displaystyle S,T\in \mathcal{J}(X,Y)}$ then ${\displaystyle S+T\in \mathcal{J}(X,Y)}$; and

(2) if ${\displaystyle W}$ and ${\displaystyle Z}$ are Banach spaces over ${\displaystyle \mathbb{K}}$ with ${\displaystyle A\in \mathcal{L}(W,X)}$ and ${\displaystyle B\in \mathcal{L}(Y,Z)}$, and if ${\displaystyle T\in \mathcal{J}(X,Y)}$, then ${\displaystyle BTA\in \mathcal{J}(W,Z)}$.