In a recent paper [11], R.E. Curto, S.H. Lee and J. Yoon asked the following question. Let A be a subnormal operator, and assume that A(2) is quasinormal. Does it follow that A is quasinormal? In this paper, we answer that question in the affirmative. In fact, we prove a more general result that subnormal nth roots of quasinormal operators are quasinormal. Research on this problem has led us to a new criterion for a semispectral measure on the half-line to be spectral, written in terms of its two "moments". (C) 2021 Elsevier Inc. All rights reserved.
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