Shift-invariant functionals on Banach sequence spaces
Volume 214 / 2013
Abstract
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.
