We introduce a new family of classes of operators which is parallel to the classes of p-paranormal operators (for p>0). They are closely related to log-hyponormal operators defined by Tanahashi, the classes A (p) and absolute p-paranormal operators defined by Furuta- Ito-Yamazaki, whose background is recent development of operator inequalities, e.g. the Furuta Inequality. So we discuss some relations among them and extend results due to Furuta-Ito- Yamazaki.
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