Counterexamples related to unbounded paranormal operators

Abstract

In this paper, we give an example of a densely defined non-closable paranormal operator. Then, we give another example of a densely defined closable paranormal operator whose closure fails to be paranormal. It is worth noticing that our first example says that the adjoint of a densely defined paranormal operator may have a trivial domain. In the end, we give an example of a densely defined closed one-to-one paranormal operator T whose adjoint is also injective and paranormal, but T is not normal.

Keywords

• Closable operator
• Paranormal operator
• Normal operator
• Hilbert space
• Fourier transform