Characterizations of positive operators via their powers

In this paper, we present new characterizations of normal and positive operators in terms of their powers. Among other things, we show that if $T^2$ is normal, $\mathcal{W}(T^{2k+1})$ lies on one side of a line passing through the origin (possibly including some points on the line) for some $k\in \mathbb{N}$, and $\mathrm{a}\mathrm{s}\mathrm{c}(T)=1$ (or $\mathrm{d}\mathrm{s}\mathrm{c}(T)=1$), then $T$ must be normal. This complements the previous result due to Putnam [28]. Furthermore, we prove that $T$ is normal (positive) if and only if $\mathrm{a}\mathrm{s}\mathrm{c}(T)=1$ and there exist coprime numbers $p,q\ge 2$ such that $T^p$ and $T^q$ are normal (positive). Finally, we also show that $T$ is positive if and only if $T^k$ is accretive for all $k\in \mathbb{N}$, which answers the question from [22] in the affirmative.