A bounded operator from ℓ^q to a Banach space is compact exactly when its coordinate shadow under all linear functionals forms a totally bounded set in ℓ^p; for Hilbert-space operators, this is equivalent to total boundedness of the joint numerical range.
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A characterization of compact operators on $\ell^p$-spaces
A bounded operator from ℓ^q to a Banach space is compact exactly when its coordinate shadow under all linear functionals forms a totally bounded set in ℓ^p; for Hilbert-space operators, this is equivalent to total boundedness of the joint numerical range.