On Strong Unboundedness of Symmetric Operators

Abstract

It will be shown that for each positive odd integer $n$ there is a symmetric operator $\mathcal{T}$ in a separable Hilbert space $\mathcal{H}$ such that $\mathcal{T},{\mathcal{T}}^3,\dots ,{\mathcal{T}}^n$ are unbounded from below and ${\mathcal{T}}^k\ge 0$ for $k>n$.