2023/06/19 by Marcoux, Laurent W., Radjavi, Heydar, Troscheit, Sascha +1 · 1 citation
#47A65. Secondary: 28A20 #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 47A45
paper · doi:10.48550/arxiv.2306.11202
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 ⊕ A is equivalent to B ⊕ 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 J2(A) of A⊕ A is assumed to be equivalent to the corresponding square root J2(B) of B ⊕ 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 \ Uα\α∈ Ω of unitary operators, no two of which are unitarily equivalent, such that each Uα is unitarily equivalent to Jn(Uα), a ``primitive" nth root of Uα⊕ Uα⊕ ⋯ ⊕ Uα.