Hyponormality of Generalized Ces\`aro Matrices of Every Positive Integer Order

Abstract

For a positive integer $m$ and a real parameter $\alpha>-1$, consider the generalized Ces\`aro matrix on $\ell^2(\mathbb N_0)$ with entries $m(n-j+1)_{m-1}/(n+\alpha+1)_m$ for $j\le n$. We give a computer-assisted proof that this operator is hyponormal if and only if $\alpha\ge0$. The main algebraic ingredient is an explicit decomposition of an auxiliary defect operator into a positive diagonal operator and at most $m$ rank-one operators. Discrete Gram polynomials determine the signs and coefficients of these terms. For nonnegative parameters, weighted estimates reduce positivity to scalar inequalities. A uniform analytic estimate covers every $m\ge32768$, while finite rational certificates cover the remaining orders and entire parameter intervals, without parameter sampling or truncating an infinite operator. The proof also gives a uniform weighted lower bound for the self-commutator. For $-1<\alpha<0$, finite-support vectors give negative quadratic forms for every order.