Yun Sung Choi

We study the denseness or norm of numerical radius attaining multilinear mappings and polynomials between Banach spaces, and examine the relations between norms and numerical radii of such mappings.

Abstract In this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following: (i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$. (ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.

We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space $X$, i.e. when the equality $$ \|\mathop{\rm Id}+P\|=1+\|P\| $$ is satisfied for all weakly compact polynomials $P:X\to X$. We show that this is the case whe

We study the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces C(K, X) and L∞(μ, X) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every σ-finite measure μ, which does not hold any more in the real case. We give an example of a complex Banach space X such that, for every k ≥ 2, the polynomial numerical index of order k of X is the

We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $\lambda_{\varphi, w}$ is the Shilov boundary for algebras of holomorphic functions on $\lambda_{\var

We obtain lower bounds for the usual norms and the absolutely summing norms of polynomials and multilinear mappings from l(p) or c(0) to a Banach space. Our estimates unify and extend the work of several previous authors who have worked with scalar-valued mappings.

Given an entire mapping $f\in \mathcal{H}_b(X,X)$ of bounded type from a Banach space $X$ into $X$, we denote by $\overline{f}$ the Aron-Berner extension of $f$ to the bidual $X^{\ast\ast}$ of $X$. We show that $\overline{g\circ f} = \overline{g}\circ \overline{f}$ for all $f, g\in \mathcal{H}_b(X,X)$ if $X$ is symmetrically regular. We also give a counterexample on $l_1$ such that the equality does not hold. We prove that the closure of the numerical range of $f$ is the same as that of $\bar{f}