On an inequality for spectral radius

In this note, we first extend and then give a related result to an inequality involing the spectral radius of nonnegative matrices that recently appcared in the literature.     

On estimate for numerical radius of some contractions

For the numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpe proved the inequality , where n ≥ 2 is the nilpotency order of the operator T. In the present paper, we prove a Haagerup-de la Harpe-type inequality for the numerical radius of contractions from more general classes. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1335–1339, October, 2006.     

Von Neumann’s inequality for noncommuting contractions

Let T₁,…,T_d be linear contractions on a complex Hilbert space and p a complex polynomial in d variables which is a sum of d single variable polynomials. We show that the operator norm of p(T₁,…,T_d) is bounded by

     

Numerical radius inequalities for operator matrices

We prove a numerical radius inequality for operator matrices, which improves an earlier inequality due to Hou and Du. As an application of this numerical radius inequality, we derive a new bound for the zeros of polynomials.     

Numerical range and quasi-sectorial contractions

A method developed in Arlinski? (1987) [1] is applied to study the numerical range of quasi-sectorial contractions and to prove three main results. Our first theorem gives characterization of the maximal sectorial generator A in terms of the corresponding contraction semigroup {exp(−tA)}_t?0. The second result establishes for these quasi-sectorial contractions a quite accurate localization of their numerical range. We give for this class of semigroups a new proof of the Euler operator-norm approximation: exp(−tA)=limn→∞(I+tA/n)⁻ⁿ, t?0, with the optimal estimate: O(1/n), of the convergence rate, which takes into account the value of the sectorial generator angle (the third result).     

Numerical radius of positive matrices

We study some properties of the numerical radius of matrices with non-negative entries, and explicit ways to compute it. We also characterize positive matrices with equal spectral and numerical radii, i.e., positive spectral matrices.     

Numerical computation of least constants for the Sobolev inequality

Summary Least constantsc for the well-known Sobolev inequality fcf _{m, G} ,fH ^m (G) are obtained in closed form by a reproducing kernel technique, where the Sobolev spaceH ^m (G) for a domainG in ⁿ is defined as the completion ofC ^m (G) with respect to the Sobolev norm given by , where is the norm ofL ₂ (G) and is the supremum norm onG. Numerical values for the case whereG is the ⁿ are given.     

A C0 interior penalty method for a fourth‐order variational inequality of the second kind

In this article, we propose a C⁰ interior penalty method for the frictional plate contact problem and derive both a priori and a posteriori error estimates. We derive an abstract error estimate in the energy norm without additional regularity assumption on the exact solution. The a priori error estimate is of optimal order whenever the solution is regular. Further, we derive a reliable and efficient a posteriori error estimator. Numerical experiments are presented to illustrate the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 36–59, 2016     

Faces for a linear inequality in 0–1 variables

Given a linear inequality in 0–1 variables we attempt to obtain the faces of the integer hull of 0–1 feasible solutions. For the given inequality we specify how faces of a variety of lower-dimensional inequalities can be raised to give full-dimensional faces. In terms of a set, called a “strong cover”, we obtain necessary and sufficient conditions for any inequality with 0–1 coefficients to be a face, and characterize different forms that the integer hull must take. In general the suggested procedures fail to produce the complete integer hull. Special subclasses of inequalities for which all faces can be generated are demonstrated. These include the “matroidal” and “graphic” inequalities, where a count on the number of such inequalities is obtained, and inequalities where all faces can be derived from lower dimensional faces.     

Nikol’skii-Stechkin inequality for trigonometric polynomials in L 0

In this paper, the Nikol’skii-Stechkin inequality for the trigonometric polynomials is generalized to the space L ₀. The resulting estimates are final.     

Numerical radius for the asymptotic stability of delay differential equations

In this note, we make use of the numerical radius to investigate the asymptotic stability of linear systems of delay differential equations. We present examples to illustrate our assumption is weaker than the classical ones.     

Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

We obtain a decomposition for multivariable Schur-class functions on the unit polydisk which, to a certain extent, is analogous to Agler's decomposition for functions from the Schur-Agler class. As a consequence, we show that d-tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d-variable von Neumann inequality for an arbitrary operator-valued bounded analytic function on the polydisk. Also, this decomposition yields a necessary condition for solvability of the finite data Nevanlinna-Pick interpolation problem in the Schur class on the unit polydisk.     

Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

C．R．Rao不等式的简单证明

引进多边矩阵理论中的分解算子,对单位阵进行正交投影分解,再利用投影矩阵秩等于迹及矩阵秩≥0的性质,给出了C.R.Rao不等式的一个简单证明.     

Modular Lattice for C 0-Operators

We prove modularity of the lattice Lat (T) of closed invariant subspaces for a C ₀-operator T with property (P). To my parents