On the reduction of powers of self-adjoint operators

Abstract

Let T ∈ B(H) be such that Tⁿ is self-adjoint for some n ∈ N with n ≥ 3. The paper’s primary aim is to establish the conditions that lead to the selfadjointness of T. We pay particular attention to the case where T³ = 0 and how it implies T is complex symmetric.

Mathematics Subject Classification (2020): Primary: 47A62; Secondary: 47B15, 47B44, 47B65, 47A05.