Abstract
A continuous linear operator on a complex Banach space is said to be paranormal if âTxâ2 †âT2xâ âxâ for all x â X. T is called totally paranormal if Tâλ is paranormal for every λ â C. In this paper we investigate the class of totally paranormal operators. We shall see that Weyl's theorem holds for operators in this class. We also show that for totally paranormal operators the Weyl spectrum satisfies the spectral mapping theorem. In Section 5 of this paper we investigate the operator equations eT = eS and eTeS = eSeT for totally paranormal operators T and S.
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