Time-Scaled Intertwining Cocycles and Identifiability of Multi-Semigroup Mixtures on Hilbert Operator Networks

Abstract

We prove that a network of dissipative semigroups $\mathcal S_i(t)=e^{-tA_i}$ admits time-scaled cocycles $K_{ij}\mathcal S_j(t)=\mathcal S_i(\lambda_{ij}t)K_{ij}$, $K_{ik}=K_{ij}K_{jk}$, if and only if the renormalized generators $\{\tau_iA_i\}$ form a common isospectral class with matching eigenspace dimensions; the scaling factors are then rigid, $\lambda_{ij}=\tau_i/\tau_j$, and eigenspaces transport isomorphically across sectors. The operators $K_{ij}$ constitute parallel transport in a flat Hilbert bundle over the index network; flatness follows from the intertwining constraints, not assumed. The mixture observable $M(t)=\sum_i w_i\mathcal B_0K_{0i}\mathcal S_i(t)\psi_i$ reduces under finite spectral support to a structured exponential sum. Under spectral separation, sector tags are uniquely recoverable; under eigenspace observability, active state components are determined. Finite-window exact reconstruction holds from $2L$ samples. The stability bound $\|\widehat\Theta-\Theta_\ast\|_{\mathcal X}\le C_{\mathrm{stab}}\kappa_{\mathrm{exp}}\varepsilon$ holds with constants explicit in the spectral geometry and observability of the network.