Time-Scaled Intertwining Cocycles and Identifiability of Multi-Semigroup Mixtures on Hilbert Operator Networks
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911532703547392 |
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| author | Alexa, Anton |
| author_facet | Alexa, Anton |
| contents | We prove that a network of dissipative semigroups $\mathcal S_i(t)=e^{-tA_i}$ admits time-scaled cocycles $K_{ij}\mathcal S_j(t)=\mathcal S_i(λ_{ij}t)K_{ij}$, $K_{ik}=K_{ij}K_{jk}$, if and only if the renormalized generators $\{τ_iA_i\}$ form a common isospectral class with matching eigenspace dimensions; the scaling factors are then rigid, $λ_{ij}=τ_i/τ_j$, 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 $M(t)=\sum_i w_i\mathcal B_0K_{0i}\mathcal S_i(t)ψ_i$ 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 $\|\widehatΘ-Θ_\ast\|_{\mathcal X}\le C_{\mathrm{stab}}κ_{\mathrm{exp}}\varepsilon$ holds with constants explicit in the spectral geometry and observability of the network. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20322 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Time-Scaled Intertwining Cocycles and Identifiability of Multi-Semigroup Mixtures on Hilbert Operator Networks Alexa, Anton Functional Analysis Operator Algebras Spectral Theory 47D06, 47A10, 47B02, 93B30 We prove that a network of dissipative semigroups $\mathcal S_i(t)=e^{-tA_i}$ admits time-scaled cocycles $K_{ij}\mathcal S_j(t)=\mathcal S_i(λ_{ij}t)K_{ij}$, $K_{ik}=K_{ij}K_{jk}$, if and only if the renormalized generators $\{τ_iA_i\}$ form a common isospectral class with matching eigenspace dimensions; the scaling factors are then rigid, $λ_{ij}=τ_i/τ_j$, 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 $M(t)=\sum_i w_i\mathcal B_0K_{0i}\mathcal S_i(t)ψ_i$ 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 $\|\widehatΘ-Θ_\ast\|_{\mathcal X}\le C_{\mathrm{stab}}κ_{\mathrm{exp}}\varepsilon$ holds with constants explicit in the spectral geometry and observability of the network. |
| title | Time-Scaled Intertwining Cocycles and Identifiability of Multi-Semigroup Mixtures on Hilbert Operator Networks |
| topic | Functional Analysis Operator Algebras Spectral Theory 47D06, 47A10, 47B02, 93B30 |
| url | https://arxiv.org/abs/2603.20322 |