Weak Schatten ideal - Imported Knowlpedia Content

Definition

Let $H$ be a Hilbert space, let $p>0$ , and write $\mu_n(T)$ for the singular values of a compact operator $T$ , arranged in nonincreasing order and counted with multiplicity. The weak Schatten ideal

\mathcal L^{p,\infty}(H)=\left\{T:\sup_{n\geq1}n^{1/p}\mu_n(T)<\infty\right\}

consists exactly of compact operators satisfying $\mu_n(T)=O(n^{-1/p})$ . The displayed supremum is a natural quasinorm. This is a two-sided operator ideal, also called the weak- $p$ or Lorentz operator ideal, and it is larger than the Schatten class $\mathcal L^p(H)$ .

Comparison with Schatten classes

For $0<q<p$ , one has

\mathcal L^q\subset\mathcal L^{p,\infty},

and $\mathcal L^p\subset\mathcal L^{p,\infty}$ . The diagonal operator on $\ell^2(\mathbb N)$ with eigenvalues $n^{-1/p}$ lies in $\mathcal L^{p,\infty}$ but not in $\mathcal L^p$ . Thus weak Schatten membership records a critical polynomial decay rate without requiring summability at the endpoint.

Role in noncommutative geometry

Weak ideals express borderline spectral dimension: an inverse or resolvent whose singular values behave like $n^{-1/p}$ is weakly $p$ -summable. At $p=1$ , suitable positive operators can support singular traces, including Dixmier-type constructions. Membership alone does not ensure that every generalized limit produces the same trace value; that stronger property is measurability in the sense of singular traces. See Lord, Sukochev, and Zanin, chapters 3 and 5.

Conventions and scope

References

S. Lord, F. Sukochev, and D. Zanin, Singular Traces: Theory and Applications, De Gruyter, 2013. Publisher record. Relevant: symmetric operator ideals, weak ideals, and singular traces.
A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-maintained text. Relevant: chapter IV, infinitesimals and Dixmier traces.