Weyl’s theorem for paranormal closed operators

Abstract

In this article, we discuss a few spectral properties of 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 and 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, the 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 non-zero isolated spectral value \(\lambda \) of T is self-adjoint and satisfies \(R(E_\lambda )=N(T-\lambda I)=N(T-\lambda I)^*\).