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

Formal definition

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

Properties and examples

Operator ideals enjoy the following nice properties.

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

References

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

Functional analysis