The entities A, B, X, Y in the title are operators, by which we mean either linear transformations on a finite-dimensional vector space (matrices) or bounded (= continuous) linear transformations on a Banach space. (All scalars will be complex numbers.) The definitions and statements below are valid in both the finite-dimensional and the infinite-dimensional cases, unless the contrary is stated. 1991 Mathematics Subject Classification 15A24, 47A10, 47A62, 47B47, 47B49, 65F15, 65F30.
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Bhatia et al. (1997) studied this question.