On the closability of class totally paranormal operators

Abstract: This article delves into the analysis of various spectral properties pertaining to totally paranormal closed operators, extending beyond the confines of boundedness and encompassing operators defined in a Hilbert space. Within this class, closed symmetric operators are included. Initially, we establish that the spectrum of such an operator is non-empty and provide a characterization of closed-range operators in terms of the spectrum. Building on these findings, we proceed to prove Weyl's theorem, demonstrating that for a densely defined closed totally paranormal operator $T$, the difference between the spectrum $\sigma(T)$ and the Weyl spectrum $\sigma_w(T)$ equals the set of all isolated eigenvalues with finite multiplicities, denoted by $\pi_{00}(T)$. In the final section, we establish the self-adjointness of the Riesz projection $E_{\mu}$ corresponding to any non-zero isolated spectral value $\mu$ of $T$. Furthermore, we show that this Riesz projection satisfies the relationships $\mathrm{ran}(E_{\mu}) = \n(T-\mu I) = \n(T-\mu I)^*$. Additionally, we demonstrate that if $T$ is a closed totally paranormal operator with a Weyl spectrum $\sigma_w(T) = {0}$, then $T$ qualifies as a compact normal operator.