Schur stability conjecture for image reconstruction operators

Counterexample to both stability assertions, with arbitrarily small instability thresholds.

Conjecture 1.1 of Banerjee and Chaudhury, arXiv:2509.06715, considers

$$P(t)=W(I-tB),\qquad R(t)=I-W+(I+tB)^{-1}(2W-I),$$

where $W$ is primitive and row-stochastic, $B$ is positive semidefinite, $e=(1,\ldots,1)^\top$, and the normalized left Perron vector $\pi$ satisfies

$$Be\le \rho(B)e,\qquad \pi^\top Be>0.$$

It asserts that both spectral radii are strictly less than one throughout $0<t<2/\rho(B)$. The conjecture does not require $B$ to be entrywise nonnegative: the preceding discussion explicitly says that nonnegativity is not essential, and Example 1.2 includes negative off-diagonal entries.

Take

$$W=\begin{pmatrix}0&1\\7/30&23/30\end{pmatrix}, \qquad B=\frac1{11}\begin{pmatrix}9&-4\\-4&3\end{pmatrix}, \qquad t=\frac45.$$

Indeed,

$$W^2= \begin{pmatrix}7/30&23/30\\161/900&739/900\end{pmatrix}>0,$$

so $W$ is primitive. Furthermore,

$$\sigma(B)=\left\{1,\frac1{11}\right\},\qquad \rho(B)=1,\qquad Be=\frac1{11}\binom5{-1}\le\binom11.$$

Thus $B$ is even positive definite. The stationary vector is

$$\pi=\frac1{37}\binom7{30}, \qquad \pi^\top Be=\frac5{407}>0,$$

and $0<4/5<2/\rho(B)=2$.

Exact rational calculation gives

$$P(4/5)= \begin{pmatrix} 16/55&43/55\\ 167/550&367/550 \end{pmatrix}, \qquad R(4/5)= \begin{pmatrix} 700/1593&545/1593\\ 253/15930&15797/15930 \end{pmatrix}.$$

Nevertheless,

$$\det(I-P(4/5))=-\frac2{1375}<0, \qquad \det(I-R(4/5))=-\frac2{2655}<0.$$

For each real two-by-two matrix $M$, its monic characteristic polynomial therefore has $\chi_M(1)<0$ and $\chi_M(x)\to+\infty$ as $x\to+\infty$. Each matrix consequently has a real eigenvalue strictly greater than one:

$$\rho(P(4/5))>1,\qquad \rho(R(4/5))>1.$$

Hence both claimed stability assertions fail simultaneously.

In fact the obstruction occurs arbitrarily close to zero. Keep the same positive definite $B$, choose any $0<s<2$, and put

v=\frac1{5-s},\qquad W_s=\begin{pmatrix}0&1\v&1-v\end{pmatrix}, \qquad \pi_s=\frac1{1+v}\binom v1.

Then $W_s$ is primitive and stochastic, $Be\le e$, and

$$\pi_s^\top Be=\frac{s}{11(6-s)}>0.$$

Direct symbolic calculation yields

$$\det(I-P_s(t)) =-\frac{t(t-s)}{11(5-s)}, \qquad \det(I-R_s(t)) =-\frac{t(t-s)}{(5-s)(t+1)(t+11)}.$$

At $t=s$, both operators already have eigenvalue one, and for every $s<t<2$, both have a real eigenvalue greater than one. Since $s$ can be arbitrarily small while $\rho(B)=1$ remains fixed, no positive stability threshold depending only on $\rho(B)$ and the dimension is valid under the conjectured assumptions. The paper's separate local per-instance stability theorem remains unaffected.