Norm or numerical radius attaining polynomials on C(K)

Abstract

Let C(K, C) be the Banach space of all complex-valued continuous functions on a compact Hausdorff space K. We study when the following statement holds: every norm attaining n-homogeneous complex polynomial on C(K, C) attains its norm at extreme points. We prove that this property is true whenever K is a compact Hausdorff space of dimension less than or equal to one. In the case of a compact metric space a characterization is obtained. As a consequence we show that, for a scattered compact Hausdorff space K, every continuous n-homogeneous complex polynomial on C(K, C) can be approximated by norm attaining ones at extreme points and also that the set of all extreme points of the unit ball of C(K, C) is a norming set for every continuous complex polynomial. Similar results can be obtained if "norm" is replaced by "numerical radius."

On boundaries on the predual of the Lorentz sequence space

“…If w is a decreasing sequence of positive numbers such that w ∈ c 0 \ l 1 , the space M w 0 coincides with the space d * (w, 1) and the results obtained in [6] generalize part of the results obtained by Moraes and Romero Grados in [11]. Choi, García, Kim and Maestre [5] proved that for K scattered and X = C(K), the subset of extreme points of the unit ball of X is a boundary for A u (B X ). Acosta [1] obtained this result for every infinite K and also gave extensions to the vector valued case.…”

Numerical radius points of a bilinear mapping from the plane with the l1-norm into itself

“…Jiménez-Sevilla and Payá [5] studied the denseness of norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces. Choi, Domingo, Kim and Maestre [6] showed that for a scattered compact Hausdorff space K, every continuous n-homogeneous polynomial on C(K : C) can be approximated by norm attaining ones at extreme points and also that the set of all extreme points of the unit ball of C(K : C) is a norming set for every continuous complex polynomial. The authors obtained similar results if "norm" is replaced by "numerical radius".…”

Numerical boundaries for some classical Banach spaces

“…If K is infinite, for the norm case, it was proved by Choi, Kim, García and Maestre (scattered case) [8] and by Acosta [1] (general case) that there is no minimal closed boundary for the space of the bounded functions from B C(K) to C(K) that are holomorphic on the open unit ball. Here we prove the same result for numerical boundaries (Theorem 5.2).…”