Nihonkai Math. J. 34 (2), 91-102, (2023)
KEYWORDS: hyponormal, spectral operator, Putnam inequality, Fuglede-Putnam theorem, 47A10, 47B20

A bounded linear operator $T$ on a Hilbert space $\mathcal{H}$ is called to be hyponormal if and only if ${T}^{*}T\ge T{T}^{*}$. We study the operator $C:={T}^{*-1}T$ for an invertible hyponormal operator $T$ and show that (i) $C$ is doubly power bounded, that is ${\mathrm{sup}}_{n\in \mathrm{\mathbb{Z}}}\Vert {C}^{n}\Vert <\infty $, so it is similar to unitary (on a Hilbert space $\mathcal{K}$) [5] and hence $C$ is a spectral operator of scalar type. (ii) For each disjoint Borel sets ${\delta}_{1},\cdots ,{\delta}_{n}\subset {S}^{1}:=\{z\in \mathrm{\u2102}:|z|=1\}$ such as ${\delta}_{1}\cup \cdots \cup {\delta}_{n}={S}^{1},T$ can be decomposed to the sum of hyponormal operators $TF\left({\delta}_{j}\right)$ where $\left\{F\right(\xb7\left)\right\}$ is a spectral measure of $C$ with $C={\int}_{[0,2\pi )}{e}^{i\theta}d{F}_{\theta}$. In particular, if $\mathfrak{R}eT:=\frac{T+{T}^{*}}{2}0$ (positive, invertible) then $C$ is similar to a unitary operator on the same Hilbert space $\mathcal{H}$. (iii) If $F\left(\delta \right)$ is self-adjoint, i.e., it is an orthogonal projection, then $F\left(\delta \right)\mathcal{H}$ is a reducing subspace of $T$. (iv) If $\lambda \in \sigma \left(C\right)$ is an isolated point of $\sigma \left(C\right)$ then $\lambda $ is an eigenvalue of $C$ and $F\left(\right\{\lambda \left\}\right)$ is self-adjoint with $F\left(\right\{\lambda \left\}\right)\mathcal{H}=\mathrm{ker}(C-\lambda )=\mathrm{ker}(C-\lambda {)}^{*}$. (v) An inequality of Putnam type for $TF\left(\delta \right)$ and (vi) If both $R=\mathfrak{R}\mathfrak{e}T$ and $S=\mathfrak{I}\mathfrak{m}T$ are positive invertible then ${R}^{-1/2}T{R}^{-1/2}$ has a spectral decomposition

where $U$ is unitary similar to $C$ and $U={\int}_{{e}^{i\theta}\in \sigma \left(U\right)}{e}^{i\theta}\u200ad{E}_{\theta}$ is its spectral decomposition.