  Unlike the uncertainty relationships of two arbitrary incompatible
observables represented by the product of variances in the past, representing
them by the sum of variances is better as it guarantees to be nontrivial for
two incompatible operators in some special cases. Although the uncertainty
relation is formulated as the sum of variances for unitary operators has been
confirmed, its general forms for arbitrary non-Hermitian operators have not
been yet investigated in detail. Thus, this study develops four sum uncertainty
relations for arbitrary non-Hermitian operators acting on system states by
utilizing an appropriate Hilbert-space metric. The compatible forms of our sum
inequalities with the conventional quantum mechanics are also provided via
$G$-metric formalism. Concrete examples demonstrate the validity of the
purposed sum uncertainty relations in both $\mathcal{PT}$-symmetric and
$\mathcal{PT}$-broken phases. The proposed methods and results can help the
reader to understand in-depth the usefulness of $G$-metric formalism in
non-Hermitian quantum mechanics and the sum uncertainty relations of
incompatible operators within.
