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 \(U_\alpha \oplus U_\alpha \oplus \cdots \oplus U_\alpha \) .
