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Concentration of mass on the Schatten classes. (English) Zbl 1115.46006

Summary: Let \(1\leq p\leq\infty\) and \(\widetilde{B(S^n_p)}\) be the unit ball of the Schatten trace class of matrices on \(\mathbb{C}^n\) or on \(\mathbb{R}^n\), normalized to have Lebesgue measure equal to one. We prove that \[ \lambda\left(\left\{T\in \widetilde{B(S^n_p)}:\frac{\|T\|_{HS}}{n} \geq c_1t\right\}\right)\leq\exp(-c_2t n^{k_p}) \] for every \(t\geq 1\), where \(k_p=\min\{2,1+p/2\}\), \(c_1,c_2>1\) are universal constants and \(\lambda\) is the Lebesgue measure. This concentration of mass inside a ball of radius proportional to \(n\) follows from an almost constant behaviour of the \(L_q\) norms (with respect to the Lebesgue measure on \(\widetilde{B(S^n_p)})\) of the Hilbert–Schmidt operator norm of \(T\). The same concentration result holds for every classical ensembles of matrices like real symmetric matrices, Hermitian matrices, symplectic matrices or antisymmetric Hermitian matrices. The result is sharp when \(p=1\) and \(p\geq 2\).

MSC:

46B07 Local theory of Banach spaces
46B28 Spaces of operators; tensor products; approximation properties
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
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