$, $ \| T + \lambda A \| \geq \| ( T + \lambda A ) e_n \| = | 1 - \frac{(1 - \lambda)}{n+1} | \longrightarrow 1 $, so that $ \| T + \lambda A \| \geq \|T\| $ for all $ \lambda $. Now for $ x \in l_1 \setminus \{0\} , $ we get $\|(T- A) x\|= \sum \limits_{n=1}\limits ^\infty |a_n| |1- \frac{2}{n+1} | <  \sum \limits_{n=1}\limits ^\infty |a_n| |1- \frac{1}{n+1}| = \|Tx\|$, which implies that $\|(T- A) x\| < \|Tx\|$ for all $x\in l_1 \setminus \{0\}$. In particular, we must have, $Ax \notin (Tx)^-$ for all $x\in l_1 \setminus \{0\}$.\
Thus, we observe that in this example, although $ T \perp_{B} A, $ it is not possible to find a sequence $ \{y_{n}\} $ in $ S_{\mathbb{X}} $ such that $ \| Ty_n \| \rightarrow \|T\| $ as $ n \rightarrow \infty $ and $Ay_n\in(Ty_n)^-$ for all $n\in \mathbb{N}$.

Next we characterize Birkhoff-James orthogonality of rotund points in the space of bounded linear operators. First we need the following lemma, which also gives a characterization of rotund points. Note that the proof of the lemma can be obtained similarly as the proof of Theorem $2.4$ 