On the closability of class totally paranormal operators

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.