Approximation by Polynomials with Nonnegative Coefficients and the Spectral Theory of Positive Operators (original) (raw)

Approximations of positive operators and continuity of the spectral radius III

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 1994

We prove estimates on the speed of convergence of the ‘peripheral eigenvalues’ (and principal eigenvectors) of a sequence Tn of positive operators on a Banach lattice E to the peripheral eigenvalues of its limit operator T on E which is positive, irreducible and such that the spectral radius r(T) of T is a Riesz point of the spectrum of T (that is, a pole of the resolvent of T with a residuum of finite rank) under some conditions on the kind of approximation of Tn to T. These results sharpen results of convergence obtained by the authors in previous papers.

Approximation by positive operators

Linear Algebra and its Applications, 1992

If A = B + iC is a normal operator, where B,C are Hermitian, then in each unitarily invariant norm, the positive part of B is a best approximation to A from the class of positive operators. This generalizes results proved earlier by P. R. Halmos, T. Ando, and R. Bouldin for special norms. Some related results are included.

Approximation theorems for certain positive linear operators

Applied Mathematics Letters, 2010

In this work we prove approximation theorems for certain positive linear operators via Ditzian-Totik moduli ω 2,φ (f , •) of second order where the step-weights are functions whose squares are concave. The results obtained are applied to the q-Lupaş-Bernstein operators, the ω, q-Bernstein operators and the convergence of the iterates of the q-Bernstein polynomials.

Approximation with restricted spectra

Mathematische Zeitschrift, 1975

The study of approximation by positive operators was initiated by Halmos, who revealed among other things that the distance 6(A) of an operator A to the set of positive operators is related to the distance of a continuous complex-valued function on the plane from the real cone of continuous nonnegative-valued functions. The analogous study of approximating a continuous complex-valued function by continuous real-valued functions leads to interesting results for selfadjoint approximation. The first part of this paper, namely Section 2, deals with approximating a bounded linear operator by self-adjoint operators. In particular, all self-adjoint best approximants to an operator T are shown to be bounded between two "extremal" operators. This result leads in turn to a classification of those operators having a unique self-adjoint best approximant. Also, those operators having the zero element (which will be denoted by "0") as a self-adjoint best approximant are characterized. These results are reminiscent of similar findings obtained in [1] and [3] for the case of positive approximation. Finally the extreme points of the convex set of best approximants to a given operator are characterized in the case that the underlying Hilbert space is finite dimensional. The second and main part of this paper, namely Section 3, is devoted to the approximation of normal operators. There we generalize the space of approximating operators from the self-adjoint operators to the operators 9~(~), namely, those normal operators whose spectra lie in some closed set Q. Since the selfadjoint operators are all normal operators whose spectra are subsets of the real line, this is indeed a generalization. The above problem was first studied in [6]. Our main result here is to classify those normal operators having a unique best approximant from 9~(~2). Finally we find necessary and sufficient conditions for a normal operator to have a diagonal best approximant from 9l(O). In order to simplify the exposition, we will employ the following conventions and notation. The symbol H will denote a complex Hilbert space. All linear operators will be assumed to be bounded linear operators on an arbitrary complex Hilbert space unless otherwise stated. B(H) will denote the set of bounded linear operators on H. We will use o-(T) to denote the spectrum of T for any TeB(H).

On the positive approximation property

Positivity, 2015

We extend and improve our earlier results on automatic regularity of continuous algebra homomorphisms between Riesz algebras of regular operators.

On the Bonsall cone spectral radius and the approximate point spectrum

Discrete and Continuous Dynamical Systems

We study the Bonsall cone spectral radius and the approximate point spectrum of (in general non-linear) positively homogeneous, bounded and supremum preserving maps, defined on a max-cone in a given normed vector lattice. We prove that the Bonsall cone spectral radius of such maps is always included in its approximate point spectrum. Moreover, the approximate point spectrum always contains a (possibly trivial) interval. Our results apply to a large class of (nonlinear) max-type operators. We also generalize a known result that the spectral radius of a positive (linear) operator on a Banach lattice is contained in the approximate point spectrum. Under additional generalized compactness type assumptions our results imply Krein-Rutman type results.

On spectral approximation. Part 1. The problem of convergence

RAIRO. Analyse numérique, 1978

L'accès aux archives de la revue « RAIRO. Analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ R.A.I.R.O. Analyse numérique/Numerical Analysis (vol. 12, n° 2, 1978, p. 97 à 112). ON SPECTRAL APPROXIMATION PARTI. THE PROBLEM OF CONVERGENCE (*) by Jean DESCLOUX (*), Nabil NASSIF (2) and Jacques RAPPAZ (X) Communiqué par P.-A. RAVIART Abstract.-One studies the problem of the numerical approximation of the spectrum of noncompact operators in Banach spaces. Special results are derived for the self adjoint case. An example is presented.