We show that control systems with an analytic semigroup and control and observation operators that are not too unbounded have a Hankel operator that belongs to the Schatten class Sp for all positive p. This implies that the Hankel singular values converge to zero faster than any polynomial rate. This in turn implies fast convergence of balanced truncations. As a corollary, decay rates for the e...

This article provides sufficient conditions for positive maps on the Schatten classes Jp; 1 p < 1 of bounded operators on a separable Hilbert space such that a corresponding Perron-Frobenius theorem holds. With applications in quantum information theory in mind sufficient conditions are given for a trace preserving, positive map on J1, the space of trace class operators, to have a unique, stric...

Let Sp be the von Neumann-Schatten ideal of compact operators in a separable Hilbert space. In the paper, upper and lower bounds for the regularized determinants of operators from Sp are established.

MICHAEL K. TIPPETTy, STEPHEN E. COHNz, RICARDO TODLINGx, AND DAN MARCHESIN{ Abstract. The Schatten p-norm condition of the discrete-time Lyapunov operator LA defined on matrices P 2 Rn n by LAP P APAT is studied for stable matrices A 2 Rn n . Bounds are obtained for the norm of LA and its inverse that depend on the spectrum, singular values and radius of stability of A. Since the solution P of ...

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...

We study non-linear traces of the Choquet type and Sugeno on algebra compact operators. They have certain partial additivities. show that these additivities characterize both respectively. There exists a close relation between majorization theory. trace class operators for type. More generally we discuss Schatten–von Neumann p-class determine when they form Banach spaces. This is an attempt at ...

Let M (ω) be the modulation space with parameters p, q and weight function ω. We prove that if p1 = p2, q1 = q2, ω1 = ω0ω and ω2 = ω0, and ∂ a/ω0 ∈ L ∞ for all α, then the Ψdo at(x,D) : M p1,q1 (ω1) → M22 (ω2) is continuous. If instead a ∈ M p,q (ω) for appropriate p, q and ω, then we prove that the map here above is continuous, and if in addition pj = qj = 2, then we prove that at(x,D) is a Sc...

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 prove the orthogonality (in the sense of Birkhoff) of the range and the kernel of an important class of elementary operators with respect to the Schatten p-class. 1. Introduction. Let H be a separable infinite-dimensional complex Hilbert space and let B(H) denote the algebra of all bounded operators on H into itself. Given A, B ∈ B(H), we define the generalized derivation δ A,B : B(H) B(H) b...