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 $\sigma(T)\setminus\omega(T)=\pi_{00}(T)$, where $\sigma(T), \omega(T)$ and $\pi_{00}(T)$ denote the spectrum, Weyl spectrum and the set of all isolated eigenvalues with finite multiplicities, respectively. Finally, we prove that the Riesz projection $E_\lambda$ with respect to any isolated spectral value $\lambda$ of $T$ is self-adjoint and satisfies $R(E_\lambda)=N(T-\lambda I)=N(T-\lambda I)^*$.