Full Length Article
Exact asymptotic volume and volume ratio of Schatten unit balls

https://doi.org/10.1016/j.jat.2020.105457Get rights and content

Abstract

The unit ball Bpn(R) of the finite-dimensional Schatten trace class Spn consists of all real n×n matrices A whose singular values s1(A),,sn(A) satisfy s1p(A)++snp(A)1, where p>0. Saint Raymond (1984) showed that the limit limnn12+1p(VolBpn(R))1n2 exists in (0,) and provided both lower and upper bounds. In this manuscript we use the theory of logarithmic potentials in external fields to determine the precise limiting constant and thus the exact asymptotic volume of Bpn(R). The corresponding result for complex Schatten balls is also obtained. As an application we compute the precise asymptotic volume ratio of the Schatten p-balls, as n, thereby extending Saint Raymond’s estimate in the case of the nuclear norm (p=1) to the full regime 1p with exact limiting behavior.

Section snippets

Proof of Theorem 1

We start with the following observation. The equality Δ()=14 was established by Saint Raymond [25, Corollaire 4 on p. 69]. For 0<p<, which is always assumed in the following, Saint Raymond [25, p. 70] characterized the constant Δ(p) as the limit of Δn(p), as n, where logΔn(p)=sup0t1tn2n(n1)1i<jnlog|titj|1plog1ni=1ntip.He then showed that the positive sequence Δn(p) is decreasing, which implies that it converges to a limit, as n, denoted by Δ(p). By providing bounds on this

Proof of Theorem 2

Let us recall some definitions and provide some additional preliminaries. Let X be a real N-dimensional Banach space with unit ball BX. If we are given a complex Banach space, we ignore the complex structure and consider the space as a real one, so that N is the dimension over R. We denote by EX the (unique) maximal volume ellipsoid that is contained in BX. The volume ratio of X is then defined as vr(X)=VolN(BX)VolN(EX)1N,where VolN() stands for the usual N-dimensional Lebesgue measure. Note

Acknowledgments

ZK has been supported by the German Research Foundation under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics - Geometry - Structure. JP has been supported by a Visiting International Professor Fellowship from the Ruhr University Bochum and its Research School PLUS and by the Austrian Science Fund (FWF) Project P32405 “Asymptotic Geometric Analysis and Applications”. ZK and CT have been supported by the DFG Scientific Network Cumulants, Concentration and

References (34)

  • BrazitikosS. et al.
  • DefantA. et al.

    Norms of tensor product identities

    Note Mat.

    (2005)
  • DefantA. et al.

    Volume estimates in spaces of homogeneous polynomials

    Math. Z.

    (2009)
  • GeissS.

    Grothendieck numbers and volume ratios of operators on Banach spaces

    Forum Math.

    (1990)
  • GordonY. et al.

    Absolutely summing operators and local unconditional structures

    Acta Math.

    (1974)
  • HiaiF. et al.
  • HoeffdingW.

    The Strong Law of Large Numbers for U-StatisticsMimeo Report 302

    (1961)
  • Cited by (12)

    • The maximum entropy principle and volumetric properties of Orlicz balls

      2021, Journal of Mathematical Analysis and Applications
    View all citing articles on Scopus
    View full text