Ranks and explicit structure of some matrices in the Quantics Tensor Train format, which allows representation with logarithmic complexity in many cases, are investigated. The matrices under consideration are Laplace operator with various boundary conditions in D dimensions and inverse Laplace operator with Dirichlet and Dirichlet-Neumann boundary conditions in one dimension. The minimal-rank e...

The property of cyclicity of a linear operator, or equivalently the property of simplicity of its spectrum, is an important spectral characteristic that appears in many problems of functional analysis and applications to mathematical physics. In this paper we study cyclicity in the context of rank-one perturbation problems for self-adjoint and unitary operators. We show that for a fixed non-zer...

We characterize possible spectra of rank-one perturbations B a self-adjoint operator A with discrete spectrum and, in particular, prove that the may include any number real or non-real eigenvalues arbitrary algebraic multiplicity

Let Γ be a finitely generated, torsion-free, two step nilpotent group. Let C(Γ) be the universal C∗-algebra of Γ. We show that acsr(C∗(Γ)) = acsr(C((Γ̂)1), where for a unital C∗-algebra A, acsr(A) is the absolute connected stable rank of A, and where (Γ̂)1 is the space of one-dimensional representations of Γ. For the case of stable rank, we have close results. In the process, we give a stable ran...

The rank of the matrix multiplication operator for n×n matrices is one of the most studied quantities in algebraic complexity theory. I prove that the rank is at least 3n2−o(n2). More precisely, for any integer p ≤ n− 1 the rank is at least (3− 1 p+1 )n2 − (1 + 2p 2p p−1 ) )n. The previous lower bound, due to Bläser, was 5 2 n2−3n (the case p = 1). The new bounds improve Bläser’s bound for all ...

The transition operator that describes the time evolution of the state probability distribution for continuousstate linear systems is given by an integral operator. A state-discretization approach is proposed, which consists of a finite rank approximation of this integral operator. As a result of the state-discretization procedure, a Markov chain is obtained, in which case the transition operat...

In this paper, we first characterize reflexive one-sided A -submodules U of a unital operator algebra A in B(H ) completely. Furthermore we investigate the invariant subspace lattice Lat R and the reflexive hull Ref R, where R is the submodule generated by rank-one operators in U ; in particular, if L is a subspace lattice, we obtain when the rank-one algebra R of Alg L is big enough to determi...

In this letter, we propose an algorithm for recovery of sparse and low rank components of matrices using an iterative method with adaptive thresholding. In each iteration, the low rank and sparse components are obtained using a thresholding operator. This algorithm is fast and can be implemented easily. We compare it with one of the most common fast methods in which the rank and sparsity are ap...

There has been much interest in so-called SIC-POVMs: rank 1 symmetric in-formationally complete positive operator valued measures. In this paper we discuss the larger class of POVMs which are symmetric and informationally complete but not necessarily rank 1. This class of POVMs is of some independent interest. In particular it includes a POVM which is closely related to the discrete Wigner func...