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 ${\displaystyle BTA\in {\mathcal {J}}(W,Z)}$ .

Operator ideals enjoy the following nice properties.

• Every component ${\displaystyle {\mathcal {J}}(X,Y)}$  of an operator ideal forms a linear subspace of ${\displaystyle {\mathcal {L}}(X,Y)}$ , 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 ${\displaystyle {\mathcal {J}}}$ , every component of the form ${\displaystyle {\mathcal {J}}(X):={\mathcal {J}}(X,X)}$  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

• Pietsch, Albrecht: Operator Ideals, Volume 16 of Mathematische Monographien, Deutscher Verlag d. Wiss., VEB, 1978.