Universal weighted sampling of positive and normal operator orbits

Abstract: We study common weighted sampling designs for operator systems that are exactly observable at nonnegative integer times. For positive operators with original frame condition number at most $κ$, we construct a common integer design with arbitrarily small relative Gramian error and $O(\log R)$ distinct times up to $R$. The optimal logarithmic counting coefficient for preserving the frame property satisfies $C<sup>*(κ)\simπ<sup>{-2}\logκ$ as $κ\to\infty$. Real and integer times give the same infimum, and we determine its asymptotics as $κ\downarrow1$. A finite bound on the sampled condition number strictly increases this coefficient. For tight frames, we determine the exact optimum under this constraint and construct an integer design attaining it. Without this constraint, the infimum is never attained. For positive operators, a design valid for every finite $κ$ requires counting faster than $\log R$, with arbitrarily slow divergence of the ratio. For unrestricted normal operators and fixed $κ&gt;1$, at least linear counting is necessary. Under the logarithmic sector condition $|\argλ|\le M(-\log|λ|)$ with fixed $M\ge0$, logarithmic sampling is possible with a sharp leading coefficient. For fixed Borel spectral sets satisfying a radial density condition, we characterize logarithmic sampling and determine the minimum upper counting exponent, attained by an integer design. Counterexamples show that the radial condition cannot be omitted.