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Essential Spectra of Induced Operators on Subspaces and Quotients

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 $T$.