A Functional Block Positivity Characterization of Normal Matrices

Abstract

A functional version of the block positivity characterization of normal matrices is established, thereby answering in a corrected form, a question posed in \cite{zhang2025more}. In particular, if \(A\in M_n\) and \(\alpha\in\mathbb{R}\setminus\{1\}\), we show that
\[
\begin{bmatrix}
(A^*A)^{\alpha} & A^*\\
A & (A^*A)^{1-\alpha}
\end{bmatrix},
\]
is positive if and only if \(A\) is normal. An operator analog of this power-type criterion on infinite-dimensional Hilbert spaces is also proved; in this setting, the corresponding conclusion is formulated in terms of semi-hyponormality.