Shift-invariant functionals on Banach sequence spaces

Albrecht Pietsch¹

¹ Biberweg 7, D-07749 Jena, Germany

Studia Mathematica, Tome 214 (2013) no. 1, pp. 37-66

Résumé

The present paper is a continuation of [23], from which we know that the theory of traces on the Marcinkiewicz operator ideal \[ \mathfrak M (H):= \bigg\{ T \in \mathfrak L (H) : \sup_{1 \le m \infty} \frac 1{\log m +1} \sum_{n=1}^m a_n(T) \infty \bigg\} \] can be reduced to the theory of shift-invariant functionals on the Banach sequence space \[ \mathfrak w (\mathbb N_0) := \bigg\{ c = (\gamma_l) : \sup_{0 \le k \infty} \frac 1{k + 1} \sum_{l=0}^k |\gamma_l| \infty \bigg\}. \]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 $\mathfrak M^\ast (H)$. Of particular interest are the sets formed by the Dixmier traces and the Connes–Dixmier traces (see [2], [4], [6], and [13]).As an intermediate step, the corresponding subspaces of $\mathfrak w^\ast (\mathbb N_0)$ are treated. This approach has a significant advantage, since non-commutative problems turn into commutative ones.

Détail

DOI : 10.4064/sm214-1-3

Keywords: present paper continuation which know theory traces marcinkiewicz operator ideal mathfrak bigg mathfrak sup infty frac log sum t infty bigg reduced theory shift invariant functionals banach sequence space mathfrak mathbb bigg gamma sup infty frac sum gamma infty bigg final purpose studies which finished following using density character measure want determine size subspaces dual mathfrak ast particular interest sets formed dixmier traces connes dixmier traces see intermediate step corresponding subspaces mathfrak ast mathbb treated approach has significant advantage since non commutative problems turn commutative