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 $𝕂\in \{ℝ,ℂ\}$, 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 $𝕂$, 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 $𝕂$ 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.