Subnormal nth roots of quasinormal operators are quasinormal

Abstract 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”.