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 {\displaystyle T} belongs to an operator ideal J {\displaystyle {\mathcal {J}}} , then for any operators A {\displaystyle A} and B {\displaystyle B} which can be composed with T {\displaystyle T} as B T A {\displaystyle BTA} , then B T A {\displaystyle BTA} is class J {\displaystyle {\mathcal {J}}} as well. Additionally, in order for J {\displaystyle {\mathcal {J}}} to be an operator ideal, it must contain the class of all finite-rank Banach space operators.
Formal definition
Let L {\displaystyle {\mathcal {L}}} denote the class of continuous linear operators acting between arbitrary Banach spaces. For any subclass J {\displaystyle {\mathcal {J}}} of L {\displaystyle {\mathcal {L}}} and any two Banach spaces X {\displaystyle X} and Y {\displaystyle Y} over the same field K ∈ { R , C } {\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}} , denote by J ( X , Y ) {\displaystyle {\mathcal {J}}(X,Y)} the set of continuous linear operators of the form T : X → Y {\displaystyle T:X\to Y} such that T ∈ J {\displaystyle T\in {\mathcal {J}}} . In this case, we say that J ( X , Y ) {\displaystyle {\mathcal {J}}(X,Y)} is a component of J {\displaystyle {\mathcal {J}}} . An operator ideal is a subclass J {\displaystyle {\mathcal {J}}} of L {\displaystyle {\mathcal {L}}} , containing every identity operator acting on a 1-dimensional Banach space, such that for any two Banach spaces X {\displaystyle X} and Y {\displaystyle Y} over the same field K {\displaystyle \mathbb {K} } , the following two conditions for J ( X , Y ) {\displaystyle {\mathcal {J}}(X,Y)} are satisfied:
(1) If S , T ∈ J ( X , Y ) {\displaystyle S,T\in {\mathcal {J}}(X,Y)} then S + T ∈ J ( X , Y ) {\displaystyle S+T\in {\mathcal {J}}(X,Y)} ; and (2) if W {\displaystyle W} and Z {\displaystyle Z} are Banach spaces over K {\displaystyle \mathbb {K} } with A ∈ L ( W , X ) {\displaystyle A\in {\mathcal {L}}(W,X)} and B ∈ L ( Y , Z ) {\displaystyle B\in {\mathcal {L}}(Y,Z)} , and if T ∈ J ( X , Y ) {\displaystyle T\in {\mathcal {J}}(X,Y)} , then B T A ∈ J ( W , Z ) {\displaystyle BTA\in {\mathcal {J}}(W,Z)} .Properties and examples
Operator ideals enjoy the following nice properties.
- Every component J ( X , Y ) {\displaystyle {\mathcal {J}}(X,Y)} of an operator ideal forms a linear subspace of L ( X , Y ) {\displaystyle {\mathcal {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 {\displaystyle {\mathcal {J}}} , every component of the form J ( X ) := J ( X , X ) {\displaystyle {\mathcal {J}}(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
- Pietsch, Albrecht: Operator Ideals, Volume 16 of Mathematische Monographien, Deutscher Verlag d. Wiss., VEB, 1978.