On the Closability of Class Totally Paranormal Operators

Abstract

This article explores 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 char-
acterization 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 σw(T) equals the set of all isolated eigenvalues with finite multiplicities, denoted by π00(T). In the final section,
we establish the self-adjointness of the Riesz projection Eμ corresponding to any non-zero isolated spectral value μ of T. Furthermore, we show that this Riesz projection satisfies the relationships ran(Eμ) = ker(T − μI) = ker(T − μI)∗. Additionally, we demonstrate that if T is a closed totally paranormal operator with a Weyl spectrum σw(T) = 0, then T qualifies as a compact normal operator.