Characterizations of positive operators via their powers
Hranislav Stanković
TL;DR
This work develops new criteria for when an operator on a complex Hilbert space is positive or normal by inspecting its powers. It generalizes Putnam’s results through conditions on the square $T^2$, the numerical range of odd powers, and ascent/descent, showing that certain power-structure constraints force normality, and that positivity can be recovered from accretivity of powers, even under relaxed hypotheses. A key contribution is the equivalence between normality (and positivity) and the existence of coprime exponent pairs with powers possessing the respective property. The results yield practical criteria for identifying positive operators and connect accretivity of powers to the spectral and numerical-range geometry, with implications for operator theory and quantum-mechanical contexts where positivity and normality are central.
Abstract
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 $\mathrm{asc\,}(T)= 1$ (or $\mathrm{dsc\,}(T)=1$), 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 $\mathrm{asc\,}(T)= 1$ and there exist coprime numbers $p,q\geq 2$ 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.