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 \(\varepsilon >0\), there exists \(\eta \) such that if \(\Vert Tx\Vert > 1 - \eta \), then there is \(x_0\) such that \(\Vert x_0 - x\Vert < \varepsilon \) 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 \(\mathcal {A}_{{{\,\mathrm{nu}\,}}}(X)\) 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 \(\mathcal {A}_{\text {nu}}(X)\) 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 \(\mathcal {A}_{{{\,\mathrm{nu}\,}}}(X)\) and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums.

在本文中，我们有兴趣研究从X到Y满足以下所有满足范数的算子T的集合\（\ mathcal {A} _ {\ Vert \ cdot \ Vert}（X，Y）\）\（\ varepsilon> 0 \），存在\（\ eta \）使得如果\（\ Vert Tx \ Vert> 1-\ eta \）则存在\（x_0 \）使得\（\ Vert x_0 -x \ Vert <\ varepsilon \），并且T本身在\（x_0 \）处达到其规范。我们证明\（c_0 \）上达到其规范的每个规范都属于\（\ mathcal {A} _ {\ Vert \ cdot \ Vert}（c_0，\ mathbb {K}）\）。此外，我们证明了类似结果既不适用于\（\ mathcal {A} _ {\ Vert \ cdot \ Vert}（\ ell _1，\ mathbb {K}）\）也不适用于\（\ mathcal {A} _ { \ Vert \ cdot \ Vert}（\ ell _ {\ infty}，\ mathbb {K}）\）。在某些假设下，我们表明压缩算子的范围属于\（\ mathcal {A} _ {\ Vert \ cdot \ Vert}（X，Y）\），并且当其中一些假设不再成立时被丢弃。还定义并研究了算子的数值半径而不是其范数的类似集合\（\ mathcal {A} _ {{{\，\ mathrm {nu} \，}}}}（X）\）。我们对属于集合\（\ mathcal {A} _ {\ Vert \ cdot \ Vert}（X，X）\）和\（\ mathcal {A} _ {\ text {nu}}（X）\），当\（X = c_0 \）或\（\ ell _ {p} \）时。结果，我们得到这些空间上的正则投影\（P_N \）属于我们的集合。我们给出属于\（\ mathcal {A} _ {\ Vert \ cdot \ Vert}（X，X）\）但不属于\（\ mathcal {A} _ {{{ \，\ mathrm {nu} \，}}}（X）\），反之亦然。最后，我们建立一些技术，使我们可以使用直接和将两个集合连接起来。