A closed densely defined operator $T$ such that both $T$ and $T^*$ are injective and paranormal yet $T$ is not normal

Simple examples of non closable paranormal operators

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.

Weyl's theorem for paranormal closed operators

In this article we discuss a few spectral properties of a paranormal closed operator (not necessarily bounded) defined in a Hilbert space. This class contains closed symmetric operators. First we show that the spectrum of such an operator is non empty. Next, we give a characterization of closed range operators in terms of the spectrum. Using these results we prove the Weyl's theorem: if $T$ is a densely defined closed, paranormal operator, then $σ(T)\setminusω(T)=π_{00}(T)$, where $σ(T), ω(T)$ and $π_{00}(T)$ denote the spectrum, Weyl spectrum and the set of all isolated eigenvalues with finit…

关于类全怪异算子的可闭性

On the closability of class totally paranormal operators

本文研究了 Hilbert 空间中全怪异闭算子的谱性质，证明了谱非空、Weyl 定理成立，并给出了 Riesz 投影的自伴性及紧正规算子的判定条件。