For 1 ≤ p < ∞ let Cp denote the usual von Neumann-Schatten ideal of compact operators on 2. The standard basis of Cp is a conditional one and so it is of interest to be able to identify the sets of coordinates for which the corresponding projection is bounded. In this paper we survey and extend the known classes of bounded projections of this type. In particular we show that some recent results...

In this paper we obtain new bounds for the minimum output entropies of random quantum channels. These bounds rely on random matrix techniques arising from free probability theory. We then revisit the counterexamples developed by Hayden and Winter to get violations of the additivity equalities for minimum output Rényi entropies. We show that random channels obtained by randomly coupling the inpu...

Singular values of a data in a matrix form provide insights on the structure of the data, the effective dimensionality, and the choice of hyper-parameters on higher-level data analysis tools. However, in many practical applications such as collaborative filtering and network analysis, we only get a partial observation. Under such scenarios, we consider the fundamental problem of recovering spec...

The Schatten quasi-norm was introduced tobridge the gap between the trace norm andrank function. However, existing algorithmsare too slow or even impractical for large-scale problems. Motivated by the equivalencerelation between the trace norm and its bilin-ear spectral penalty, we define two tractableSchatten norms, i.e. the bi-trace and tri-tracenorms, and prov...

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related a conjecture an open question were presented by R. Lemos G. Soares in \cite{lemos}. In addition, we present complement unitarily invariant norm inequality was conjectured Bhatia, Y. Lim T. Yamazaki \cite{Bhatia2}, recently proved T.H. Dinh...

The goal of this review is to explain some recent results [5] regarding generalizations of the Stein-Tomas (and Strichartz) inequalities to the context of trace ideals (Schatten spaces).

2 Definitions and main results 3 2.1 Notations and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Graph norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Schatten-von Neumann classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Hölder and weakly Hölder graphs . . . . . . . . . . . . ....

Suppose that f is a Lipschitz function on R with ‖f‖Lip ≤ 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let p ∈ (1,∞) and suppose that x ∈ B(H) is an operator such that the commutator [A, x] is contained in the Schatten class Sp. It is proved by the last two authors, that then also [f(A), x] ∈ Sp and there exists a constant Cp independent of x and f such that ‖[f(A), x]‖p ≤ ...