Stability relations for Hilbert space operators and a problem of Kaplansky

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 $\{ U_α\}_{α\in Ω}$ of unitary operators, no two of which are unitarily equivalent, such that each $U_α$ is unitarily equivalent to $J_n(U_α)$, a primitive" $n^{th}$ root of $U_{\alpha }\oplus U_{\alpha }\oplus \cdots \oplus U_{\alpha }$.