Polarization Constant for the Numerical Radius

arXiv (Cornell University), 2012

We derive bounds and asymptotics for the maximum Riesz polarization quantity M p n (A) := max x 1 ,x 2 ,... ,x n ∈A min x∈A n j=1 1 |x − x j | p (which is n times the Chebyshev constant) for quite general sets A ⊂ R m with special focus on the unit sphere and unit ball. We combine elementary averaging arguments with potential theoretic tools to formulate and prove our results. We also give a discrete version of the recent result of Hardin, Kendall, and Saff which solves the Riesz polarization problem for the case when A is the unit circle and p > 0, as well as provide an independent proof of their result for p = 4 that exploits classical polynomial inequalities and yields new estimates. Furthermore, we raise some challenging conjectures.

International Journal of Pure and Apllied Mathematics, 2013

Let A and B be unital complex Banach algebras, and ϕ be a unital surjective numerical radius preserving linear map from A into B. We discuss a Nagasawa type theorem for this maps and show that ϕ is a Jordan isomorphism, if A and B are commutative.

2005

The open problem of determining the exact value of the n-th linear polarization constant c n of R n has received considerable attention over the past few years. This paper makes a contribution to the subject by providing a new lower bound on the value of sup y =1 | x 1 , y • • • x n , y |, where x 1 ,. .. , x n are unit vectors in R n. The new estimate is given in terms of the eigenvalues of the Gram matrix [ x i , x j ] and improves upon earlier estimates of this kind. However, the intriguing conjecture c n = n n/2 remains open.

arXiv (Cornell University), 2012

We derive bounds and asymptotics for the maximum Riesz polarization quantity (which is n times the Chebyshev constant ) for quite general sets A ⊂ R m with special focus on the unit sphere and unit ball. We combine elementary averaging arguments with potential theoretic tools to formulate and prove our results. We also give a discrete version of the recent result of Hardin, Kendall, and Saff which solves the Riesz polarization problem for the case when A is the unit circle and p > 0, as well as provide an independent proof of their result for p = 4 that exploits classical polynomial inequalities and yields new estimates. Furthermore, we raise some challenging conjectures.

Proceedings of the American Mathematical Society, 2014

Let T 1 , … , T n T_1,\dots ,T_n be bounded linear operators on a complex Hilbert space H H . We study the question whether it is possible to find a unit vector x ∈ H x\in H such that | ⟨ T j x , x ⟩ | |\langle T_jx,x\rangle | is large for all j j . Thus we are looking for a generalization of a well-known fact for n = 1 n=1 that the numerical radius w ( T ) w(T) of a single operator T T satisfies w ( T ) ≥ ‖ T ‖ / 2 w(T)\ge \|T\|/2 .

Bulletin of The London Mathematical Society, 1999

We prove that if all the rank-one bounded operators on a Banach space X attain their numerical radii, then X must be reflexive, but the converse does not hold. In fact, every reflexive space with basis can be renormed in such a way that there is a rank-one operator not attaining the numerical radius.

Vector Measures, Integration and Related Topics, 2009

Let 1 ≤ p, q < ∞ and let X be a complex Banach space. For each f (z) = P ∞ n=0 xnz n with f H ∞ (D,X) ≤ 1 we define Rp,q(f, X) = sup{r ≥ 0 : x0 p + (P ∞ n=1 xn r n) q ≤ 1} and denote the Bohr radius of X by Rp,q(X) = inf{Rp,q(f, X) : f H ∞ (D,X) ≤ 1}. The aim of this note is to study for which spaces X = L s (µ) or X = s one has Rp,q(X) > 0.

Studia Mathematica, 2007

We give several sharp inequalities involving powers of the numerical radii and the usual operator norms of Hilbert space operators. These inequalities, which are based on some classical convexity inequalities for nonnegative real numbers and some operator inequalities, generalize earlier numerical radius inequalities.

2010

Bulletin of the Australian Mathematical Society

Let X be a real Banach space. The rectangular constant $\mu (X)$ and some generalisations of it, $\mu _p(X)$ for $p \geq 1$ , were introduced by Gastinel and Joly around half a century ago. In this paper we make precise some characterisations of inner product spaces by using $\mu _p(X)$ , correcting some statements appearing in the literature, and extend to $\mu _p(X)$ some characterisations of uniformly nonsquare spaces, known only for $\mu (X)$ . We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning $\mu (l_p)$ and $\mu _p(l_p)$ .