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

Abstract

We study rigidity phenomena for time-scaled intertwining families of dissipative semigroups $\mathcal S_i(t)=e^{-tA_i}$ and prove that a network of bounded injective operators satisfying $K_{ij}\mathcal S_j(t)=\mathcal S_i(\lambda_{ij}t)K_{ij}$ and $K_{ik}=K_{ij}K_{jk}$ necessarily admits a multiplicative gauge representation $\lambda_{ij}=\tau_i/\tau_j$, if and only if the renormalized generators $\{\tau_iA_i\}$ form a common isospectral class with matching eigenspace dimensions; in particular, eigenspaces are transported isomorphically across sectors. The operators $K_{ij}$ define parallel transport in a flat Hilbert bundle over the index network, with flatness derived from the intertwining constraints rather than assumed. As an application, 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, the modal parameters are uniquely identifiable, with sector tags determined intrinsically by the operator spectra; under eigenspace observability, active state components are uniquely recovered. Finite-window exact reconstruction holds from $2L$ samples, and the stability bound $\|\widehat\Theta-\Theta_\ast\|_{\mathcal X}\le C_{\mathrm{stab}}\kappa_{\mathrm{exp}}\varepsilon$ follows with constants explicitly controlled by the spectral geometry and observability of the network.