Simple examples of non closable paranormal operators

Abstract

In this note, 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. These two examples are simpler than those which first appeared in [1]. It is worth noticing that our first example tells us that the adjoint of a densely defined paranormal operator may have a trivial domain.

Cited by in corpus (2)

• Unbounded operators: (square) roots, nilpotence, closability and some related invertibility results
• A closed densely defined operator $T$ such that both $T$ and $T^{\ast }$ are injective and paranormal yet $T$ is not normal