On the closability of class totally paranormal operators
M. H. M. Rashid
TL;DR
This work develops spectral theory for densely defined closed totally paranormal operators on Hilbert spaces, extending known results from bounded to unbounded cases. It establishes that the spectrum is non-empty, provides a spectral criterion for closed range, and proves Weyl's theorem in this unbounded setting via $\sigma(T)\setminus\sigma_w(T)=\pi_{00}(T)$, while also showing that nonzero isolated spectral values have self-adjoint Riesz projections with $\mathrm{ran}(E_{\mu})=\ker(T-\mu I)=\ker(T-\mu I)^*$. Furthermore, it demonstrates that when $\sigma_w(T)=\{0\}$, the operator is compact and normal, linking spectral decompositions to finite-dimensional eigenspaces through $E_{\mu}$. The results extend and unify several prior findings for hyponormal and bounded totally paranormal operators, providing tools for spectral analysis of a broader class of unbounded operators in Hilbert spaces.
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 $σ(T)$ and the Weyl spectrum $σ_w(T)$ equals the set of all isolated eigenvalues with finite multiplicities, denoted by $π_{00}(T)$. In the final section, we establish the self-adjointness of the Riesz projection $E_μ$ corresponding to any non-zero isolated spectral value $μ$ of $T$. Furthermore, we show that this Riesz projection satisfies the relationships $\mathrm{ran}(E_μ) = \n(T-μI) = \n(T-μI)^*$. Additionally, we demonstrate that if $T$ is a closed totally paranormal operator with a Weyl spectrum $σ_w(T) = {0}$, then $T$ qualifies as a compact normal operator.