A note on numerical radius attaining mappings

We prove that if every bounded linear operator (or $N$-homogeneous polynomials) with the compact approximation property attains its numerical radius, then $X$ is a finite dimensional space. Moreover, we present an improvement of the polynomial James' theorem for numerical radius proved by Acosta, Becerra Guerrero and Gal$\'a$n in 2003. Finally, the denseness of weakly (uniformly) continuous $2$-homogeneous polynomials on a Banach space whose Aron-Berner extensions attain their numerical radii is obtained.Comment: 15 page

A generalized ACK structure and the denseness of norm attaining operators

Inspired by the recent work of Cascales et al., we introduce a generalized concept of ACK structure on Banach spaces. Using this property, which we call by the quasi-ACK structure, we are able to extend known universal properties on range spaces concerning the density of norm attaining operators. We provide sufficient conditions for quasi-ACK structure of spaces and results on the stability of quasi-ACK structure. As a consequence, we present new examples satisfying the (Lindenstrauss) property B$^k$, which have not been known previously. We also prove that property B$^k$ is stable under injective tensor products in certain cases. Moreover, ACK structure of some Banach spaces of vector-valued holomorphic functions is also discussed, leading to new examples of universal BPB range spaces for certain operator ideals.Comment: 17 page

The Daugavet and Delta-constants of points in Banach spaces

We introduce two new notions called the Daugavet constant and $\Delta$-constant of a point, which measure quantitatively how far the point is from being Daugavet point and $\Delta$-point and allow us to study Daugavet and $\Delta$-points in Banach spaces from a quantitative viewpoint. We show that these notions can be viewed as a localized version of certain global estimations of Daugavet and diametral local diameter two properties such as Daugavet indices of thickness. As an intriguing example, we present the existence of a Banach space $X$ in which all points on the unit sphere have positive $\Delta$-constants despite the Daugavet indices of thickness of $X$ being zero. Moreover, using the Daugavet and $\Delta$-constants of points in the unit sphere, we describe the existence of almost Daugavet and $\Delta$-points as well as the set of denting points of the unit ball. We also present exact values of the Daugavet and $\Delta$-constant on several classical Banach spaces, as well as Lipschitz-free spaces. In particular, it is shown that there is a Lipschitz-free space with a $\Delta$-point which is the furthest away from being a Daugavet point. Finally, we provide some related stability results concerning the Daugavet and $\Delta$-constant.Comment: 35 pages, 3 figure

Norm-attaining operators which satisfy a Bollobás type theorem

This is a pre-print of an article published in Banach Journal of Mathematical Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s43037-020-00113-7In this paper, we are interested in studying the set A(parallel to center dot parallel to) (X, Y) of all norm-attaining operators T from X into Y satisfying the following: given epsilon > 0, there exists eta such that if parallel to Tx parallel to > 1 - eta, then there is x(0) such that parallel to x(0) - x parallel to < epsilon and T itself attains its norm at x(0). We show that every norm one functional on c(0) which attains its norm belongs to A(parallel to center dot parallel to) (c(0), K). Also, we prove that the analogous result holds neither for A(parallel to center dot parallel to) (l(1), K) nor A(parallel to center dot parallel to) (l(infinity), K). Under some assumptions, we show that the sphere of the compact operators belongs to A(parallel to center dot parallel to) (X, Y) and that this is no longer true when some of these hypotheses are dropped. The analogous set A(nu)(X) for numerical radius of an operator instead of its norm is also defined and studied. We present a complete characterization for the diagonal operators which belong to the sets A(parallel to center dot parallel to) (X, X) and A(nu)(X) when X = c(0) or l(p). As a consequence, we get that the canonical projections P-N on these spaces belong to our sets. We give examples of operators on infinite dimensional Banach spaces which belong to A(parallel to center dot parallel to) (X, X) but not to A(nu)(X) and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums

The Bishop-Phelps-Bollobás properties in complex Hilbert space

In this paper, we consider theBishop–Phelps–Bollobás point propertyfor variousclasses of operators on complex Hilbert spaces, which is a stronger property thanthe Bishop–Phelps–Bollobás property. We also deal with analogous problem byreplacing the norm of an operator with its numerical radius

On the existence of non-norm-attaining operators

In this paper we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal{L}(E, F)$. By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set $K$ of $\mathcal{L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm closed convex hull, then we can guarantee the existence of an operator which does not attain its norm. This allows us to provide the following generalization of results due to Holub and Mujica. If $E$ is a reflexive space, $F$ is an arbitrary Banach space, and the pair $(E, F)$ has the bounded compact approximation property, then the following are equivalent: (i) $\mathcal{K}(E, F) = \mathcal{L}(E, F)$; (ii) Every operator from $E$ into $F$ attains its norm; (iii) $(\mathcal{L}(E,F), \tau_c)^* = (\mathcal{L}(E, F), \| \cdot \|)^*$; where $\tau_c$ denotes the topology of compact convergence. We conclude the paper presenting a characterization of the Schur property in terms of norm-attaining operators