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Operator ideal

From Wikipedia, the free encyclopedia

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 belongs to an operator ideal , then for any operators and which can be composed with as , then is class as well. Additionally, in order for to be an operator ideal, it must contain the class of all finite-rank Banach space operators.

Formal definition

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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

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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.

References

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  • Pietsch, Albrecht: Operator Ideals, Volume 16 of Mathematische Monographien, Deutscher Verlag d. Wiss., VEB, 1978.