It is well known that the power of a weakly compact operator is compact. Is the spectrum of a weakly compact operator is the same as a compact operator?
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If you consider operators on a Hilbert space, then the weak and weak-∗ topologies agree, so that the unit ball is weakly compact. This implies that every bounded operator is weakly compact. In particular, any compact subset of $\mathbb C$ can be the spectrum of a weakly compact operator.