As a further generalization of paranormal operators,

  we shall introduce a new class ``absolute-\((p,r)\)-paranormal''

  operators for \(p > 0\) and \(r > 0\) such that

  \(\||T|^p |T^*|^r x\|^r \ge \||T^*|^r x\|^{p+r}\)

  for every unit vector \(x\).

  And we shall show several properties on

  absolute-\((p,r)\)-paranormal operators

  as generalizations of the results on

  absolute-\(k\)-paranormal and \(p\)-paranormal operators

  introduced in \cite{FIY} and \cite{FIN}, respectively.