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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 finite multiplicities, respectively. Finally, we prove that the Riesz projection $E_λ$ with respect to any isolated spectral value $λ$ of $T$ is self-adjoint and satisfies $R(E_λ)=N(T-λI)=N(T-λI)^*$.