Certain properties involving the unbounded operators p(T), TT⁎, and T⁎T; and some applications to powers and nth roots of unbounded operators

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

“…Similarly, there are everywhere defined unclosable operators T such that $$T^2=I$$ on all of H , and hence $$\sigma (T^2)=\lbrace 1\rbrace$$. These examples may be consulted in [17].…”

“…As it is known, if A is a closed operator, then A 2 need not be closed. There are known counterexamples, see, for example, [17]. On the other hand, there are closable operators having closed squares, for example, if $$\mathcal {F}_0$$ denotes the restriction of the $$L^2(\mathbb {R})$$‐Fourier transform to the dense subspace $$C_0^{\infty }(\mathbb {R})$$ (the space of infinitely differentiable functions with compact support), then it is known that $$$$\begin{equation*} D(\mathcal {F}_0^2)=\lbrace 0\rbrace .$$…”

“…\end{equation*}$$Hence, $$\mathcal {F}_0$$ is unclosed while $$\mathcal {F}_0^2$$ is trivially closed on $$\lbrace 0\rbrace$$. In fact, there is an unclosable operator A , which is everywhere defined and such that $$A^2=0$$ on all of H , see again [17].…”