Approximation Properties of Locally Convex Spaces and the Problem of Uniqueness of the Trace of a Linear Operator

Details Approximation Properties of Locally Convex Spaces and the Problem of Uniqueness of the Trace of a Linear Operator Approximation Properties of Locally Convex Spaces and the Problem of Uniqueness of the Trace of a Linear Operator

2011-08-08

arxiv.org/abs/1108.1721v1

Selecta Mathematica Sovietica, vol. 11, No.1 (1992), p. 25-40

Mathematics

Functional Analysis

18 pages

Scientific paper

In the present article, it is proved that every nuclear operator in a locally convex space E has a well-defi?ned trace if E possesses the approximation property. However, even if a space possesses the approximation property this still does not guarantee a positive solution of A. Grothendieck's uniqueness problem for this space. Below, we present an example of a quasi-complete space with the approximation property in which it is not possible to de?fine the trace for all Fredholm operators (in the sense of A. Grothendieck). We prove that the uniqueness problem has a positive solution if E possesses the "bounded approximation property." Preliminary information and results are presented in Section 2. A number of approximation-type properties of locally convex spaces and relations between these properties are considered in Section 3. The principal results of the present study, along with certain corollaries from these results (for example, the existence of a matrix trace), may be found in Section 4.

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