  Let $X$ be a complex Banach space and let $T$ be a bounded linear operator on
$X$. For any closed $T$-invariant subspace $F$ of $X$, $T$ induces operators
$T_{|F}:F \longrightarrow F$ and $T/F:X/F\longrightarrow X/F$. In this note, we
give a simple proof of the fact that the essential spectra of $T_{|F}$ and
$T/F$ are always contained in the polynomial hull of the essential spectrum of
$T$.
