New subclasses of the class of $\mathcal H$-matrices and related bounds for the inverses

The paper introduces new subclasses, called $\mathrm P\mathcal H\mathrm N(\pi)$ and $\mathrm P\mathcal H\mathrm{QN}(\pi)$, of (nonsingular) $\mathcal H$-matrices of order $n$ dependent on a partition $\pi$ of the index set $\{1,\dots,n\}$, which generalize the classes $\mathrm P\mathcal H(\pi)$, introduced previously, and contain, in particular, such subclasses as those of strictly diagonally dominant (SDD), Nekrasov, $S$-SDD, $S$-Nekrasov, $\mathrm{QN}$, and $\mathrm P\mathcal H(\pi)$ matrices. Properties of the matrices introduced are studied, and upper bounds on their inverses in $l_\infty$ norm are obtained. Block generalizations of the classes $\mathrm P\mathcal H\mathrm N(\pi)$ and $\mathrm P\mathcal H\mathrm{QN}(\pi)$ in the sense of Robert are considered.
Also a general approach to defining subclasses $\mathcal K^\pi$ of the class $\mathcal H$ containing a given subclass $\mathcal{K\subset H}$ and dependent on a partition $\pi$ is presented.