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A real valued function f defined on a subset of R is δ2-ward continuous if limn→∞Δ³ f(Xn) = 0 whenever limn→∞ >Δ³ Xn = 0, where Δ³ zn -- Zn + 3 - 3zn+2 + 3zn+i - Zn for each positive integer n, R denotes the set of real numbers, and a subset E of R is d²-ward compact if any sequence of points in E has a d²-quasi Cauchy subsequence where a sequence (xn) is d²-quasi Cauchy if limn→∞Δ³zn=0. It turns out that the uniform limit process preserves this kind of continuity, and the set of d²-ward continuous functions is a closed subset of the set of continuous functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the absolute-(p; r) -*-paranormal which is a further generalization of both absolute-k-*-paranormal operator and p-*-paranormal operator. Also, we introduce the (p; r; q)-*-paranormal operator, which is also a further generalization of both absolute-(p; r) -*-paranormal operator and p -*-paranormal operator. We will show ba- sic structural properties of these two classes of operators. Finally, we will see some properties of the p -*-paranormal operator. Some spectral properties are also presented: we show that, if T is p-*-paranormal then σjp(T) = σp(T), and σja(T) = σa(T). [ABSTRACT FROM AUTHOR]