Operator ideal

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