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In this paper we study some properties of a totally -paranormal operator (defined below) on Hilbert space. In particular, we characterize a totally -paranormal operator. Also we show that Weyl’s theorem and the spectral mapping theorem hold for totally -paranormal operators through the local spectral theory. Finally, we show that every totally -paranormal operator satisfies an analogue of the single valued extension property for and some of totally -paranormal operators have scalar extensions....
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