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For a completely non-unitary contraction T, some necessary (and, in certain cases, sufficient) conditions are found for the range of the $H^{\infty }$ calculus, $H^{\infty }\left(T\right)$, and the commutant, T’, to contain non-zero compact operators, and for the finite rank operators of T’ to be dense in the set of compact operators of T’. A sufficient condition is given for T’ to contain non-zero operators from the Schatten-von Neumann classes $S_p$.