In this paper, we consider the stability of an inverse spectral problem for a nonsymmetric ordinary differential operator. We give an estimate for deviation in the coefficients of this operator when the spectral data perturb. Our result shows that if two spectral data are sufficiently close to one another, then the corresponding two differential operators must be close each other in the sense o...

We show that given an estimate  that is close to a general high-rank positive semidefinite (PSD) matrix A in spectral norm (i.e., ‖Â−A‖2 ≤ δ), the simple truncated Singular Value Decomposition of  produces a multiplicative approximation of A in Frobenius norm. This observation leads to many interesting results on general high-rank matrix estimation problems: 1. High-rank matrix completion: we...

We obtain an explicit asymptotic formula for the norms of the spectral projections of the non-self-adjoint harmonic oscillator H. We deduce that the spectral expansion of e−Ht is norm convergent if and only if t is greater than a certain explicit positive constant.

For a given family of matrices F , we present a methodology to construct (complex) polytope Barabanov norms (and, when F is nonnegative, polytope Barabanov monotone norms and antinorms). Invariant Barabanov norms have been introduced by Barabanov and constitute an important instrument to analyze the joint spectral radius of a set of matrices. In particular, they played a key role in the disprov...

The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm always exists. This approach lends itself easily to the analysis of further properties of the generali...

We study a simple two step procedure for estimating sparse precision matrices from data with missing values, which is tractable in high-dimensions and does not require imputation of the missing values. We provide rates of convergence for this estimator in the spectral norm, Frobenius norm and element-wise `∞ norm. Simulation studies show that this estimator compares favorably with the EM algori...

We study a simple two step procedure for estimating sparse precision matrices from data with missing values, which is tractable in high-dimensions and does not require imputation of the missing values. We provide rates of convergence for this estimator in the spectral norm, Frobenius norm and element-wise `∞ norm. Simulation studies show that this estimator compares favorably with the EM algori...

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of FredholmVolterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L∞ norm and weighted L2-norm. The numerical examp...

In the recent years, the trace norm of graphs has been extensively studied under the name of graph energy. The trace norm is just one of the Ky Fan k-norms, given by the sum of the k largest singular values, which are studied more generally in the present paper. Several relations to chromatic number, spectral radius, spread, and to other fundamental parameters are outlined. Some results are ext...

We prove that for a right linear bounded normal operator on a quaternionic Hilbert space (quaternionic bounded normal operator) the norm and the numerical radius are equal. As a consequence of this result we give a new proof of the known fact that a non zero quaternionic compact normal operator has a non zero right eigenvalue. Using this we give a new proof of the spectral theorem for quaternio...