Operator ideals and approximation properties of Banach spaces Seminar

After a short introduction to the language of Pietsch's famous theory of operator ideals (including some illustrative examples), we will concentrate on a deepening of the relationship between (maximal) adjoint Banach ideals and conjugate Banach ideals. This leads to somehow surprising links between the existence of a suitable operator ideal norm on operator ideal products (which in general do not inherit an ideal norm from their normed factors), Grothendieck-approximation properties of Banach spaces, a transfer of the famous Principle of Local Reflexivity of Lindenstrauss, Rosenthal, Johnson and Zippin to operator ideals, and some "nice" factorisation properties of finite rank operators, formulated in the language of so called "accessible operator ideals". For example, we will recognise that also the maximal Banach ideal of absolutely 1-summing operators (and its dual) does not satisfy Floret's "m.a.p. factorization property".

Our aim is to give a rather algebraic approach by - occasionally - linking the language of general operator ideals with the no less important and powerful language of Grothendieck's tensor norms on tensor products of Banach spaces and trace duality, leading to further interesting results between certain operator ideal components, their dual spaces and approximation of finite rank operators in those operator ideal components.

Finally, we will present a few open problems.