Formal definition
Let denote the class of continuous linear operators acting between arbitrary Banach spaces. For any subclass of and any two Banach spaces and over the same field , denote by the set of continuous linear operators of the form such that . In this case, we say that is a component of . An operator ideal is a subclass of , containing every identity operator acting on a 1-dimensional Banach space, such that for any two Banach spaces and over the same field , the following two conditions for are satisfied:
- (1) If then ; and
- (2) if and are Banach spaces over with and , and if , then .
Properties and examples
Operator ideals enjoy the following nice properties.
- Every component of an operator ideal forms a linear subspace of , 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 , every component of the form 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.