A bounded linear operator $T$ from a Banach space $X$ to a Banach space $Y$ is called norm-attaining, if there exists a vector $x\in X$ with $\|x\|=1$ such that $$\|Tx\|=\|T\|.$$ Let $\mathbb{D}=\{z\in \mathbb{C}: |z|<1\}$. Consider the disc algebra $$A(\mathbb{D}):=\{f\in\text{ Hol}(\mathbb{D}):f \text{ is continuous on }\overline{\mathbb{D}},$$ $$\|f\|_{\infty}=\sup_{z\in\mathbb{D}}|f(z)|<\infty\}.$$

Can we get a concrete example of non-norm-attaining bounded linear operator $T: A(\mathbb{D}) \longrightarrow A(\mathbb{D})$?