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

Formal definition

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

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

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

Properties and examples

Operator ideals enjoy the following nice properties.

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