Suppose that we observe entries or, more generally, linear combinations of entries of an unknown m×T -matrix A corrupted by noise. We are particularly interested in the high-dimensional setting where the numbermT of unknown entries can be much larger than the sample size N . Motivated by several applications, we consider estimation of matrix A under the assumption that it has small rank. This c...

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...

We address some theoretical guarantees for Schatten-p quasi-norm minimization (p ∈ (0, 1]) in recovering low-rank matrices from compressed linear measurements. Firstly, using null space properties of the measuring operator, we provide a sufficient condition for exact recovery of low-rank matrices. This condition guarantees unique recovery of matrices of ranks equal or larger than what is guaran...

Let H(S) be the Hardy space on the unit sphere S in C, n ≥ 2. Consider the Hankel operator Hf = (1 − P )Mf |H(S), where the symbol function f is allowed to be arbitrary in L(S, dσ). We show that for p > 2n, Hf is in the Schatten class Cp if and only if f − Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if ...

We study when multiplication by a weight can turn non-compact composition operator on \(H^2\) into compact operator, and it be in Schatten classes. The \(q\)-summing case \(H^p\) is considered. also this one.

Let E and P be nonnegative quadratic forms in a Hilbert space H and assume that E + bP is densely defined and closed for every b ≥ 0. For every b > 0 let Hb be the self-adjoint operator associated with E + bP in the sense of Kato’s representation theorem. By Kato’s monotone convergence theorem, the operators (Hb + 1) −1 converge strongly to an operator L, as b tends to infinity. Let k ∈ N. We g...

Let H denote a separable, infinite-dimensional complex Hilbert space. A two-sided ideal / of operators on H possesses the generalized Fuglede property (GFP) if, for every normal operator N and every X e L(H), NX XN e I implies N*X XN* e /. Fuglede's Theorem says that / = {0} has the GFP. It is known that the class of compact operators and the class of Hilbert-Schmidt operators have the GFP. We ...

This is a survey article. We consider different problems in connection with the behavior of functions of operators under perturbations of operators. We deal with three classes of operators: unitary operators, self-adjoint operators, and contractions. We study operator Lipschitz and operator differentiable functions. We also study the behavior of functions under perturbations of an operator by a...

The quantum speed limit is a fundamental upper bound on the speed of quantum evolution.However, the actualmathematical expression of this fundamental limit depends on the choice of ameasure of distinguishability of quantum states.We show that quantum speed limits are qualitatively governed by the Schatten-p-normof the generator of quantumdynamics. Since computing Schatten-p-norms can bemathemat...

We show that many invariant subspaces M for d-shifts (S1, . . . , Sd) of finite rank have the property that the orthogonal projection PM onto M satisfies PMSk − SkPM ∈ L, 1 ≤ k ≤ d for every p > 2d, L denoting the Schatten-von Neumann class of all compact operators having p-summable singular value lists. In such cases, the d tuple of operators T̄ = (T1, . . . , Td) obtained by compressing (S1, ....