On the numerical radius and its applications

We give a brief account of the numerical radius of a linear bounded operator on a Hilbert space and some of its better-known properties. Both finite- and infinite- dimensional aspects are discussed, as well as applications to stability theory of finite-difference approximations for hyperbolic initial-value problems.     

On some fixed-point theorems for generalized contractions and their perturbations

The aim of this paper is to prove a collection of fixed-point theorems for mappings which can be roughly called generalized contractions or their perturbations. In particular, we are going to consider operators (single-valued or multi-valued) in Banach spaces with a quasimodulus, in hyperconvex subsets of normed spaces, or finally in non-Archimedean spaces. A particular attention will be paid to Krasnoselskii-type fixed-point theorems as well as to a Schaefer-type fixed-point theorem. Some applications to nonlinear functional-integral equations will be given. Our results extend and complement some commonly known theorems.     

On the error estimate for calculating some singular integrals

We consider some classes of multi-dimensional singular integral operators with Calderon-Zygmund kernels, construct their discrete analogues and give the error estimate for approximation of continual singular integral operator by the discrete ones. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convex numerical radius

In this work we introduce the concept of convex numerical radius for a continuous and linear operator in a Banach space, which generalizes that of the classical numerical radius. Besides studying some of its properties, we give a version of James's sup theorem in terms of convex numerical radius attaining operators.     

On the norm and numerical radius of Hermitian elements

Translated from Lietuvos Matematikos Rinkinys, Vol. 34, No. 2, pp. 248–254, April–June, 1994.     

On the numerical radius of operators in Lebesgue spaces

We show that the absolute numerical index of the space Lp(μ) is (where ). In other words, we prove that

     

On the distance spectral radius of some graphs

For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. In this paper, we determine the unique graph with minimum distance spectral radius among all connected graphs of order n with a given diameter. Moreover, we determine the unique graph with maximum distance spectral radius among the catacondensed hexagonal systems with h hexagons.     

On the numerical radius of Lipschitz operators in Banach spaces

We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and C  -rich subspaces have Lipschitz numerical index 1. Moreover, using the Gâteaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we prove that the Lipschitz numerical index has the stability properties for the _c0$c_0$-, _l1$l_1$-, and _l∞$l_{\infty }$-sums of spaces and vector-valued function spaces. From this, we show that the C(K)$C(K)$ spaces, _L1(μ)$L_1(\mu )$-spaces and _L∞(ν)$L_{\infty }(\nu )$-spaces have Lipschitz numerical index 1.     

An algorithm for computing the numerical radius

Received on 23 October 1995. Revised on 15 July 1996. This paper is concerned with the calculation of the numericalradius of a matrix, an important quantity in the analysis ofconvergence of iterative processes. An algorithm is developedwhich enables the numerical radius to be obtained to a givenprecision, using a process which successively refines lowerand upper bounds. It uses an iteration procedure analogous tothe power method for computing the largest modulus eigenvalueof a Hermitian matrix. In contrast to that method, convergenceis possible here to a local maximum of the underlying optimizationproblem which is not global, so that only a lower bound is provided.This is used in conjunction with a technique based on the solutionof a generalized cigenvalue problem to provide an upper bound.Numerical results illustrate the performance of the method.     

On some topological properties of numerical algorithms

A result quantity in a numerical algorithm is considered as a function of the input data, roundoff and truncation errors. In order to investigate this functional relationship using the methods of mathematical analysis a structural model of the numerical algorithm calledR-automaton is introduced. It is shown that the functional dependence defined by anR-automaton is a continuous rational function in a neighborhood of any data point except in a point set, the Lebesgue measure of which is zero. An effective general-purpose algorithm is presented to compute the derivative of any result quantity with respect to the individual roundoff and truncation errors. Some ways of generalizing theR-automation model without losing the results achieved are finally suggested.     

Isometries of a generalized numerical radius

For 0<q<1, the q-numerical range is defined on the algebra M_n of all n×n complex matrices by

W_q(A)={x^∗Ay:x,y∈Cⁿ,∥x∥=∥y∥=1,〈y,x〉=q}.     

The numerical radius of exterior powers

Let λ₁ and α₁ be respectively the eigenvalue of largest modulus and largest singular value of a linear operator A. Then A is called radial if |λ₁| = α₁. This paper is concerned with an examination of radial compound matrices. It turns out that the radial property for compound matrices is equivalent to an investigation of the case of equality in the classical inequalities of H. Weyl relating products of eigenvalues and singular values.