Norm attaining operators which satisfy a Bollobás type theorem

Abstract

In this paper, we are interested in studying the set ${\mathcal{A}}_{\Vert \cdot \Vert }(X,Y)$ of all norm-attaining operators $T$ from $X$ into $Y$ satisfying the following: given $\epsilon >0$, there exists $\eta$ such that if $\Vert Tx\Vert >1-\eta$, then there is $x_0$ such that $\Vert x_0-x\Vert <\epsilon$ 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 ${\mathcal{A}}_{\Vert \cdot \Vert }(c_0,\mathbb{K})$. Also, we prove that the analogous result holds neither for ${\mathcal{A}}_{\Vert \cdot \Vert }({\ell }_1,\mathbb{K})$ nor ${\mathcal{A}}_{\Vert \cdot \Vert }({\ell }_{\infty },\mathbb{K})$. Under some assumptions, we show that the sphere of the compact operators belongs to ${\mathcal{A}}_{\Vert \cdot \Vert }(X,Y)$ and that this is no longer true when some of these hypotheses are dropped. The analogous set ${\mathcal{A}}_{nu}(X)$ 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 ${\mathcal{A}}_{\Vert \cdot \Vert }(X,X)$ and ${\mathcal{A}}_{nu}(X)$ when $X=c_0$ or ${\ell }_p$. 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 ${\mathcal{A}}_{\Vert \cdot \Vert }(X,X)$ but not to ${\mathcal{A}}_{nu}(X)$ and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums.

In this version, we have added a complete characterization for the sets ${\mathcal{A}}_{\Vert \cdot \Vert }$ and ${\mathcal{A}}_{nu}$ for the diagonal operators on $c_0$ and ${\ell }_p$