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What is "local theory of Banach spaces"?

Albrecht Pietsch — 1999

Studia Mathematica

Banach space theory splits into several subtheories. On the one hand, there are an isometric and an isomorphic part; on the other hand, we speak of global and local aspects. While the concepts of isometry and isomorphy are clear, everybody seems to have its own interpretation of what "local theory" means. In this essay we analyze this situation and propose rigorous definitions, which are based on new concepts of local representability of operators.

Bad properties of the Bernstein numbers

Albrecht Pietsch — 2008

Studia Mathematica

We show that the classes p b e r n : = T : ( b ₙ ( T ) ) ∈ l p associated with the Bernstein numbers bₙ fail to be operator ideals. Moreover, p b e r n ∘ q b e r n ⊈ r b e r n for 1/r = 1/p + 1/q.

Shift-invariant functionals on Banach sequence spaces

Albrecht Pietsch — 2013

Studia Mathematica

The present paper is a continuation of [23], from which we know that the theory of traces on the Marcinkiewicz operator ideal ( H ) : = T ∈ ( H ) : s u p 1 ≤ m < ∞ 1 / ( l o g m + 1 ) ∑ n = 1 m a ₙ ( T ) < ∞ can be reduced to the theory of shift-invariant functionals on the Banach sequence space ( ℕ ₀ ) : = c = ( γ l ) : s u p 0 ≤ k < ∞ 1 / ( k + 1 ) ∑ l = 0 k | γ l | < ∞ . The final purpose of my studies, which will be finished in [24], is the following. Using the density character as a measure, I want to determine the size of some subspaces of the dual *(H). Of particular interest are the sets formed by the Dixmier traces and the Connes-Dixmier traces...

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