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 B T A , then B T A 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.
Let L denote the class of continuous linear operators acting between two Banach spaces. For any subclass J of L and any two Banach spaces X and Y over the same field K , denote by J ( X , Y ) the set of continuous linear operators of the form T : X → Y . 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 ∈ J ( X , Y ) then S + T ∈ J ( X , Y ) ; and (2) if W and Z are Banach spaces over K with A ∈ L ( W , X ) and B ∈ L ( Y , Z ) , and if T ∈ J ( X , Y ) , then B T A ∈ 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 operatorsWeakly compact operatorsFinitely strictly singular operatorsStrictly singular operatorsCompletely continuous operators