On the closability of paranormal operators (Q624590)

The author considers the closability of paranormal operators. A linear operator \(A:D(A)\subset\mathcal H \to\mathcal H\), where \(\mathcal H\) is a Hilbert space and \(D(\cdot)\) denotes the domain, is said to be paranormal if \(||Ax||^2 \leq ||x||\, ||A^2x||\) for all \(x\in D(A^2)\). Evidently, hyponormal operators (i.e., \(A\) satisfying \(A^*A-AA^*\geq 0\)) are paranormal. Paranormal operators share many properties with hyponormal operators. It is known that hyponormal operators are closable and the closure of a hyponormal operator is also hyponormal. However, the questions of the closability and the properties of the closure of a paranormal operator were open until now. The author of this paper gives negative answers to both questions.