Density and Spectral Criteria for Universal Nonuniform Sampling of Positive Operator Orbits

Abstract

We characterize locally finite sets \(T\subset[0,\infty)\) that preserve exact observability from nonnegative integer times for every bounded positive operator \(A\) and bounded observation operator \(C\). The Stein identity reduces universality to uniform upper and lower bounds on radial Stein--Pick cones over all finite spectral models. Writing \(N_T(R)=\#(T\cap[0,R])\), we prove that universal Bessel preservation is equivalent to \(N_T(R)=O(R)\). The conditions \(0\in T\) and \(N_T(R)\asymp R\) characterize universal observability preservation under \([A,C^*C]=0\), as well as uniform bound preservation for each fixed number of spectral values. However, we construct a clustered set satisfying these conditions whose optimal sampled lower bounds tend to zero as the spectral count grows, despite fixed integer observability bounds. The same set destroys the frame property of a singly generated positive diagonal Carleson frame while preserving its Bessel property. When natural density exists, universality is equivalent to \(0\in T\) and \(0<d(T)<\infty\). More generally, containing zero and a subset of positive finite natural density suffices under linear upper counting. For sets containing zero, universality is invariant under sublinear differences of counting functions. For general normal operators, universal Bessel preservation is instead equivalent to uniformly bounded counts on unit intervals; under a fixed logarithmic sector condition, it is again characterized by linear upper counting.