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

In this paper, we are interested in studying the set ${\mathcal{A}}_{\mathrm{\parallel }\cdot \mathrm{\parallel }}(X,Y)$ of all norm-attaining operators $T$ from $X$ into $Y$ satisfying the following: given $ϵ>0$, there exists $\eta$ such that if $\mathrm{\parallel }Tx\mathrm{\parallel }>1-\eta$, then there is $x_0$ such that $| x_0 - x| < \epsilon$and$T$itselfattainsitsnormat$x_0$. We show that every norm one functional on $c_0$ which attains its norm belongs to ${\mathcal{A}}_{\mathrm{\parallel }\cdot \mathrm{\parallel }}(c_0,\mathbb{K})$. Also, we prove that the analogous result holds neither for ${\mathcal{A}}_{\mathrm{\parallel }\cdot \mathrm{\parallel }}({\mathrm{\ell }}_1,\mathbb{K})$ nor ${\mathcal{A}}_{\mathrm{\parallel }\cdot \mathrm{\parallel }}({\mathrm{\ell }}_{\mathrm{\infty }},\mathbb{K})$. Under some assumptions, we show that the sphere of the compact operators belongs to ${\mathcal{A}}_{\mathrm{\parallel }\cdot \mathrm{\parallel }}(X,Y)$ 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 ${\mathcal{A}}_{\mathrm{\parallel }\cdot \mathrm{\parallel }}(X,X)$ and ${\mathcal{A}}_{nu}(X)$ when $X=c_0$ or ${\mathrm{\ell }}_p$. 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 ${\mathcal{A}}_{\mathrm{\parallel }\cdot \mathrm{\parallel }}(X,X)$ 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.