For questions about operator ideals.
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 $ \mathcal J $, then for any operators $ A $ and $ B $ which can be composed with $ T $ as $ B T A $ is class $ \mathcal J $ as well. Additionally, in order for $ \mathcal J $ to be an operator ideal, it must contain the class of all finite-rank Banach space operators.
Let $ \mathcal L $ denote the class of continuous linear operators acting between arbitrary Banach spaces. For any subclass $ \mathcal J $ of $ \mathcal L $ and any two Banach spaces $ X $ and $ Y $ over the same field $ { \mathbb K } \in \left\lbrace { \mathbb R } , { \mathbb C } \right\rbrace $, denote by $ \mathcal J ( X , Y ) $ the set of continuous linear operators of the form $ T : X \to Y $ such that $ T \in \mathcal J $. In this case, we say that $ \mathcal J ( X , Y ) $ is a component of $ \mathcal J $. An operator ideal is a subclass $ \mathcal J $ of $ \mathcal 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 $ \mathbb K $, the following two conditions for $ \mathcal J ( X , Y ) $ are satisfied:
- If $ S , T \in \mathcal J ( X , Y ) $ then $ S + T \in \mathcal J ( X , Y ) $; and
- if $ W $ and $ Z $ are Banach spaces over $ \mathbb K $ with $ A \in \mathcal L ( W , X ) $ and $ B \in \mathcal L ( Y , Z ) $, and if $ T \in \mathcal J ( X , Y ) $, then $ B T A \in \mathcal J ( W , Z ) $.
Operator ideals enjoy the following nice properties.
- Every component $ \mathcal J ( X , Y ) $ of an operator ideal forms a linear subspace of $ \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 $ \mathcal J $, every component of the form $ \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
Source: Wikipedia