Show that there is no operator $T \in L(\ell^2(\mathbb{N}))$ Let $S \in L(\ell^2(\mathbb{N}))$ be the unilateral shift. How do we show that if $K \in L(\ell^2(\mathbb{N}))$ is a compact operator, then there is no operator $T \in L(\ell^2(\mathbb{N}))$ such that $T^2=S^3 +K$?

"Compact perturbation" should always make one consider "Fredholm". And this is the key here: if we consider the Fredholm index, noting that $S^3+K$ is Fredholm we would have 
$$
2\operatorname{ind}(T)=\operatorname{ind}(T^2)=\operatorname{ind}(S^3+K)=\operatorname{ind}(S^3)=-3. 
$$
This would require $\operatorname{ind}(T)=-\tfrac32$, which is impossible. 
Note that $T$ is necessarily Fredholm, because we have 
$$
[(S^*)^3T]T=I+K',\ \ \ T[T(S^*)^3=I+K''
$$
for certain compact $K',K''$. 
