Numerical radius inequalities and its applications in estimation of zeros of polynomials

Abstract

We present some upper and lower bounds for the numerical radius of a bounded linear operator defined on complex Hilbert space, which improves on the existing upper and lower bounds. We also present an upper bound for the spectral radius of sum of product of n pairs of operators. As an application of the results obtained, we provide a better estimation for the zeros of a given polynomial.2010 Mathematics Subject Classification. Primary 47A12, 15A60, 26C10.

Bounds for zeros of a polynomial using numerical radius of Hilbert space operators

“…Bhunia et al [5] proved that where C = C(p). Carmichael and Mason [11] proved that As an application of the results obtained in Sect.…”

“…To mention a few of them are Cauchy [11], Fujii and Kubo [9], Alpin et al [4], Kittaneh [14], Linden [15]. One of the important techniques to obtain bounds for the zeros of the polynomial p(z) is to obtain bounds for the numerical radius of the Frobenius companion matrix C(p) of p(z), where Using the numerical radius inequalities of the Frobenius companion matrix of a given monic polynomial, Abu-Omar and Kittaneh [1], M. Al-Dolat et al [3], Bhunia et al [5] obtained various bounds for the zeros of that polynomial. We here obtain bounds for the zeros of the polynomial p(z) and give examples to show that they are better than the existing ones.…”

“…is equivalent to the operator norm. Various numerical radius inequalities improving (1.1) have been studied in [5,10,12,[16][17][18][19]. Kittaneh [13] improved on the inequality (1.1) to prove that…”

On Numerical Radius Bounds Involving Generalized Aluthge Transform

“…The study of the numerical radius of an operator defined on the Hilbert space is in the focus of researchers in these days in studying perturbation, convergence, iterative solution methods, and integrative methods, etc, see [1][2][3][4][5][6][7][8][9]. In this regard, the numerical radius inequality stated in (3) is studied extensively by various mathematicians, see [10][11][12][13][14][15][16][17][18][19][20][21]. Actually, it is interesting for the researchers to get refinements and generalizations of this inequality [22][23][24][25][26][27].…”

Refinements of A-numerical radius inequalities and their applications

“…Over the years, various numerical radius inequalities have been obtained to improve on the inequality (1.1). Interested readers can look into [1,3,4,5,10,11] and the references therein for more information on recent advances in numerical radius inequalities. In this paper, we establish some new inequalities for the numerical radius of bounded linear operators.…”