Marcinkiewicz–Macaev ideal - Imported Knowlpedia Content

Definition

Let $H$ be a separable Hilbert space, and let $\mu_n(T)$ be the singular values of a compact operator $T$ , in nonincreasing order. The Marcinkiewicz–Macaev ideal is

\mathcal M_{1,\infty}(H) =\left\{T:\sup_{N\geq 1}\frac{1}{\log(1+N)} \sum_{n=1}^{N}\mu_n(T)<\infty\right\}.

The displayed supremum defines an equivalent standard ideal norm. This logarithmic Marcinkiewicz ideal, often denoted $\mathcal L^{(1,\infty)}$ , is a two-sided symmetric operator ideal. Its condition controls Cesàro-type partial sums rather than each individual singular value.

Comparison with the weak Schatten endpoint

The weak Schatten ideal at $p=1$ is

\mathcal L^{1,\infty}_{\mathrm{weak}} =\{T:\sup_{n\geq1}n\mu_n(T)<\infty\}.

It embeds continuously in $\mathcal M_{1,\infty}$ , since $\sum_{n\leq N}n^{-1}=O(\log N)$ . The inclusion is strict: logarithmic control of an accumulated sum does not impose a uniform $O(1/n)$ bound on every singular value. Consequently, notation such as $\mathcal L^{1,\infty}$ is unsafe unless the author states whether it means weak- $\ell^1$ decay or the larger logarithmic Marcinkiewicz ideal.

The distinction is explicit in Lord, Sedaev, and Sukochev, who formulate the logarithmic ideal through partial sums of singular values.

Singular traces

For $T\geq0$ in $\mathcal M_{1,\infty}$ , the logarithmic means

\frac{1}{\log(1+N)}\sum_{n=1}^{N}\mu_n(T)

form a bounded sequence. Applying a suitably invariant generalized limit gives a Dixmier trace. This trace vanishes on trace-class operators and can detect the coefficient of critical logarithmic divergence. Membership in the ideal does not by itself make the value independent of the chosen generalized limit; that independence is an additional measurability property.

Conventions and scope

The names “Macaev ideal,” “Dixmier ideal,” and “weak trace ideal” are not used uniformly. Some sources reserve the Macaev notation for a Köthe-dual ideal, while noncommutative-geometry sources commonly use $\mathcal L^{(1,\infty)}$ for the logarithmic ideal defined here. This knowl fixes the partial-sum convention. More general Marcinkiewicz ideals replace $\log(1+N)$ by another increasing concave control function.

References

S. Lord, A. Sedaev, and F. Sukochev, “Dixmier Traces as Singular Symmetric Functionals and Applications to Measurable Operators,” Journal of Functional Analysis 224 (2005), 72–106. DOI record. Relevant: the logarithmic ideal $\mathcal L^{(1,\infty)}$ , Marcinkiewicz operator spaces, and measurable operators.
S. Lord, F. Sukochev, and D. Zanin, Singular Traces: Theory and Applications, De Gruyter, 2013. Publisher record. Relevant: chapters 3 and 5 on symmetric operator ideals and Dixmier traces.
A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-maintained text. Relevant: Chapter IV, §2 on infinitesimals of order one and the Dixmier trace.