Operator model and a trace formula for pairs of unitary operators

What the math gives to ML

The paper gives an operator-theoretic representation of pairs of unitary dynamics through matrix-valued Caratheodory functions and positive reproducing kernels. The transferable assets are exact norm preservation, structured block factorizations, and finite-dimensional positive-semidefiniteness tests for frequency responses. A practical neural-network adaptation is to use paired unitary state transitions in recurrent or state-space models, and to regularize their transfer functions toward positive realness. These mechanisms target long-horizon stability and extrapolation rather than merely adding another generic spectral penalty.