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

Formal definition

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

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

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

Properties and examples

Operator ideals enjoy the following nice properties.

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