Not from homework, but practice problems for an exam.
I am not sure how to approach this problem, it seems like it should be pretty simple. Let $X \neq \{0\}$ denote a complex normed vector space, and assume that the operator $T : D(T) ⊂ X → X $ is closed. Let $λ ∈ C$. Show that the operator $T − λI$ is closed.
Let $(x_n)$ be a sequence in $D(T)$ such that $x_n \to x$ and $y_n:=(T - \lambda I)x_n \to y$.
We have to show that $x \in D(T)$ and $(T - \lambda I)x=y$.
From $Tx_n = y_n+ \lambda x_n \to y + \lambda x$, we see, since $T$ is closed, that $x \in D(T)$ and $Tx=y + \lambda x$, hence $(T - \lambda I)x=y$.