Compact operator and its norm attaining set.

Question

Let $X=l_{2}$ and $T:X\to X$ be a compact operator. Define $M=\{x\in l_{2}: \|x\|=1 \ \text{and} \ \|Tx\|=\|T\| \}$. My question is that define $T$ such that $M$ is not compact. We know that in the case of a bounded linear operator the identity operator works as an example since in $l_{2}$ unit ball is not compact (in this case, $M$ is the unit ball).