Weyl’s theorem for paranormal closed operators

Bala, Neeru and G, Ramesh (2020) Weyl’s theorem for paranormal closed operators. Annals of Functional Analysis. ISSN 2008-8752 (In Press)

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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 σ(T) \ ω(T) = π00(T) , where σ(T),ω(T) and π00(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 Eλ with respect to any non-zero isolated spectral value λ of T is self-adjoint and satisfies R(Eλ) = N(T- λI) = N(T- λI) ∗.

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