Complete functional calculus bounds for $ρ$-contractions

What the math gives to ML

The paper gives an intrinsic positive-semidefinite characterization of ρ-contractions and sharp complete functional-calculus bounds that remain valid for matrix-valued analytic functions. The transferable asset is a dilation-based certificate: every power of a Cρ operator is a compressed unitary power, yielding a uniform long-horizon bound ∥T^n∥ ≤ ρ rather than merely controlling one-step spectral radius. This suggests replacing unconstrained recurrent or state-space transition matrices with trainable operators satisfying the Cρ positivity inequality, and using the complete matrix-valued nature of the result to obtain stability under channel or block amplification.