Let \(\mathcal{L}(X)\) be the algebra of all bounded linear operators on a complex Banach space \(X\). An operator \(T\in\mathcal{L}(X)\) is said to be {normaloid} if its spectral radius equals \(\| T\| \). The operator \(T\) is said to be {hereditarily normaloid} if every part of \(T\) is normaloid (here ``a part of \(T\)'' means ``its restriction to one of its closed invariant subspaces''), and is {totally hereditarily normaloid} if it is hereditarily normaloid and if every invertible part of \(T\) has a normaloid inverse. The class \(THN\) of totally hereditarily normaloid operators, introduced by \textit{S. V. Djordjević} and the author [Math. Proc. R. Ir. Acad. 104A, 75--81 (2004; Zbl 1089.47005)], lies properly between the classes of paranormal and normaloid operators; see the recent paper by the author, \textit{S. V. Djordjević} and \textit{C. S. Kubrusly} [Acta Sci. Math. 71, No. 1--2, 337--352 (2005; Zbl 1106.47016)]. An operator \(T\in\mathcal{L}(X)\) is said to satisfy {property \textbf{H}\((q)\)} provided that \[ H_0(T-\lambda):=\{x\in X:\lim_{n\to+\infty}\| (T-\lambda)^nx\| ^{\frac{1}{n}}=0\}=\ker(T-\lambda)^q \] for all \(\lambda\in\mathbb{C}\) and some integer \(q\geq1\). The class of operators satisfying this property will be also denoted by \textbf{H}\((q)\). It contains, amongst others, the classes consisting of generalized scalar, subscalar and totally paranormal operators on a Banach space, multipliers of semi-simple Banach algebras, and hyponormal, \(p\)-hyponormal \((0