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 ${\mathcal{S}}_i(t)=e^{-t{A}_i}$ and prove that a network of bounded injective operators satisfying $K_{ij}{\mathcal{S}}_j(t)={\mathcal{S}}_i(\hat{I}{»}_{ij}t)K_{ij}$ and $K_{ik}=K_{ij}{K}_{jk}$ necessarily admits a multiplicative gauge representation $\hat{I}{»}_{ij}=\ddot{I}{„}_i/\ddot{I}{„}_j$, if and only if the renormalized generators $\{\ddot{I}{„}_i{A}_i\}$ 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 $M(t)={\sum }_i{w}_i{\mathcal{B}}_0{K}_{0i}{\mathcal{S}}_i(t)\ddot{I}{ˆ}_i$ 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 $\Vert \hat{\hat{I}}˜-\hat{I}{˜}_{\ast }{\Vert }_{\mathcal{X}}\le C_{\mathrm{stab}}\hat{I}{º}_{\exp }\epsilon$ follows with constants explicitly controlled by the spectral geometry and observability of the network.

20 pages. Published in Complex Analysis and Operator Theory