  In his monograph on Infinite Abelian Groups, I. Kaplansky raised three ``test
problems" concerning their structure and multiplicity. As noted by Azoff, these
problems make sense for any category admitting a direct sum operation. Here, we
are interested in the operator theoretic version of Kaplansky's second problem
which asks: if $A$ and $B$ are operators on an infinite-dimensional, separable
Hilbert space and $A \oplus A$ is equivalent to $B \oplus B$ in some (precise)
sense, is $A$ equivalent to $B$? We examine this problem under a strengthening
of the hypothesis, where a ``primitive" square root $J_2(A)$ of $A\oplus A$ is
assumed to be equivalent to the corresponding square root $J_2(B)$ of $B \oplus
B$. When ``equivalence" refers to similarity of operators and $A$ is a compact
operator, we deduce from this stronger hypothesis that $A$ and $B$ are similar.
We exhibit a counterexample (due to J. Bell) of this phenomenon in the setting
of unital rings. Also, we exhibit an uncountable family $\{ U_\alpha\}_{\alpha
\in \Omega}$ of unitary operators, no two of which are unitarily equivalent,
such that each $U_\alpha$ is unitarily equivalent to $J_n(U_\alpha)$, a
``primitive" $n^{th}$ root of $U_\alpha \oplus U_\alpha \oplus \cdots \oplus
U_\alpha$.
