Inertia-Sensitive Kreiss Bounds for $J$-Selfadjoint Matrices

Let $A\in {\mathbb{C}}^{n\times n}$ have spectrum in the closed unit disk. Its maximal power growth $\mathrm{Power}(A):={\sup }_{k\ge 0}\mathrm{\parallel }{A}^k\mathrm{\parallel }$ measures transient amplification, whereas the Kreiss constant $\mathrm{Kreiss}(A):={\sup }_{\mathrm{\mid }z\mathrm{\mid }>1}(\mathrm{\mid }z\mathrm{\mid }-1)\mathrm{\parallel }(zI-A{)}^{-1}\mathrm{\parallel }$ measures the corresponding resolvent growth outside the disk. The classical finite-dimensional Kreiss theorem gives $\mathrm{Power}(A)\le en\mathrm{Kreiss}(A)$, and the linear dependence on $n$ is unavoidable for general matrices. We show that, for matrices selfadjoint with respect to an indefinite metric, the ambient dimension $n$ can be replaced by an effective dimension determined by the minimal polynomial and the inertia of the metric. Specifically, if $A^{\ast }J=JA$, where $J$ is a fundamental symmetry with inertia $(n-q,q)$, then $\mathrm{Power}(A)\le e\min \{d(A),2q+1,2(n-q)+1\}\mathrm{Kreiss}(A)$, where $d(A)$ is the degree of the minimal polynomial. Our proof requires no assumption on diagonalizability or reality on the spectrum. Instead, we associate each cyclic orbit with a finite-rank selfadjoint Hankel operator and transfer its rank and inertia to a coefficient estimate. Examples based on scaled nilpotent shifts show that the linear dependence on the smaller inertia index is asymptotically sharp, even when this index is negligible relative to the matrix size and $d(A)=n$. We also obtain scaled-disk decay estimates and weighted-norm extensions to arbitrary nonsingular Hermitian metrics.