On nth roots of bounded and unbounded quasinormal operators

Abstract

In a recent paper (JFA 278:108342, 2020), R. E. Curto, S. H. Lee and J. Yoon asked the following question: LetTbe a subnormal operator, and assume that$$T^2$$ T 2 is quasinormal. Does it follow thatTis quasinormal? In (JFA 280:109001, 2021) we answered this question in the affirmative. In the present paper, we will extend this result in two directions. Namely, we prove that hyponormal (or even much beyond this class) nth roots of bounded quasinormal operators are quasinormal. On the other hand, we show that subnormal nth roots of unbounded quasinormal operators are quasinormal. We also prove that a non-normal quasinormal operator having a quasinormal nth root has a non-quasinormal nth root.

Unbounded operators having self‐adjoint, subnormal, or hyponormal powers

“…For example, we show that if T is a quasinormal (unbounded) operator such that $$T^n$$ is normal for some $$n\ge 2$$, then T must be normal. By a recent result by Pietrzycki–Stochel in [22], we deduce that a closed subnormal operator such that $$T^n$$ is normal for some n , is necessarily normal. These results are closely related to others, which have been of some interest recently (see [2] and [21]).…”

“…We are now in a position to state and prove a result on quasinormal 𝑛th roots of normal operators (see [22] for a related result). Theorem 3.12.…”

“…We are now in a position to state and prove a result on quasinormal n th roots of normal operators (see [22] for a related result). Theorem Let T be a quasinormal (unbounded) operator such that $$T^n$$ is normal for some $$n\ge 2$$.…”

“…Proof Since $$T^n$$ is normal, it is quasinormal. By Theorem 1.4 in [22], we know that T must be quasinormal. Theorem 3.12 then gives the normality of T .$$\Box$$…”