Let $$0<p,q\le \infty $$ and denote by $$\mathscr {S}_p^N$$ {S}_q^N$$ the corresponding Schatten classes of real $$N\times N$$ matrices. We study Gelfand numbers natural identities {S}_p^N\hookrightarrow \mathscr between prove asymptotically sharp bounds up to constants only depending on p q. This extends classical results for finite-dimensional $$\ell _p$$ sequence spaces E. Gluskin non-commut...

We provide a new proof of Volberg’s Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-2 class p...

Let ‖ · ‖ be a unitary similarity invariant norm on the set Mn of n× n complex matrices. A description is obtained for surjective maps φ on Mn satisfying ‖AB− BA‖ = ‖φ(A)φ(B)−φ(B)φ(A)‖ for all A,B ∈ Mn . The general theorem covers the special cases when the norm is one of the Schatten p -norms, the Ky-Fan k -norms, or the k -numerical radii. Mathematics subject classification (2000): Primary 15...

We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we ...

Additive commutators of operators belonging to two{sided ideals of B(H) are studied. J){commutators is characterized and found to equal IJ ; B(H)]. It is shown that three commutators are enough in I; B(H)], and that four are enough in I; J ]. Finally, some results about single commutators are proved, for example that every element of the Schatten class C p is a single (C p ; B(H)){commutator if...

It will be shown that for 1 < p < 2 the Schatten p-class is isometrically isomor-phic to a subspace of the predual of a von Neumann algebra. Similar results hold for non-commutative L p (N;)-spaces deened by a semi-nite, normal, faithful trace on a von Neumann algebra N. The embeddings rely on a suitable notion of p-stable processes in the non-commutative setting.

The metric increasing property of the exponential map is known to be equivalent to the fact that the set of positive definite matrices is a Riemannian manifold of nonpositive curvature. We show that this property is an easy consequence of the logarithmic-geometric mean inequality for positive numbers. Operator versions of this inequality lead to a generalisation of the exponential metric increa...

We show convergence of the weak dual greedy algorithm in wide class of Banach spaces, extending our previous result where it was shown to converge in subspaces of quotients of Lp (for 1 < p < ∞). In particular, we show it converges in the Schatten ideals Sp when 1 < p < ∞ and in any Banach lattice which is p-convex and q-concave with constants one, where 1 < p < q < ∞. We also discuss convergen...

We correct a formula of Gavish and Donoho for singular value shrinkage with operator norm loss non-square matrices. also observe that in the classical regime, optimal any Schatten converges to best linear predictor.

We give a Coulson integral formula and Coulson-Jacobs for the p-Schatten energy of bipartite graphs. use these formulas to compare different trees by using quasiorder, establishing maximality paths minimality stars among all trees.