Norm attaining operators which satisfy a Bollob\'as type theorem

In this paper we are interested to study the set $\mathcal{A}_{\|\cdot\|}$ of all norm one linear operators $T$ from $X$ into $Y$ which attain the norm and satisfy the following: given $\epsilon>0$, there exists $\eta$, which depends on $\epsilon$ and $T$, such that if $\|T(x)\| > 1 - \eta$, then there is $x_0$ such that $\| x_0 - x\| < \epsilon$ and $T$ itself attains the norm at $x_0$. We show that every norm one functional on $c_0$ which attains the norm belongs to $\mathcal{A} (c_0, \mathbb{K})$. Also, we prove that the analogous result is not true neither for $\mathcal{A}_{\|\cdot\|} (\ell_1, \mathbb{K})$ nor $A (\ell_{\infty}, \mathbb{K})$. Under some assumptions, we show that the sphere of the compact operators belongs to $\mathcal{A}_{\|\cdot\|}$ and that this is no longer true when some of these hypotheses are dropped. We show also that the natural projections on $c_0$ or $\ell_p$, for $1\leq p < \infty$, always belong to this set. The analogous set $\mathcal{A}_{nu}$ for numerical radius of an operator instead of its norm is also defined and studied. We give non trivial examples of operators on infinite dimensional Banach spaces which belong to $\mathcal{A}_{\| \cdot \|}$ but not to $\mathcal{A}_{nu}$ and vice-versa. Finally, we establish some techniques which allow to connect both sets.