Operator-Valued Hardy Spaces and Kramers--Kronig Relations for Non-Markovian Quantum Memory Kernels

Abstract

Retarded support, upper-half-plane holomorphy, and Hardy boundary control are distinct properties of a memory kernel. We give sufficient conditions linking them for the Nakajima--Zwanzig kernel of an open system with finite-dimensional system Liouville space. If the projected kernel has an absolutely continuous real-axis representation with density w in L1 intersect Lp0, no singular part, and p0 > 1, its transform lies in the operator-valued Hardy class Hp(B) for 1 < p <= p0. A finite-dimensional componentwise argument yields the corresponding principal-value and once-subtracted Kramers--Kronig (KK) boundary formulas; the H1 endpoint is separate. Microscopic unitary evolution gives holomorphy and a trace-norm bound for reduced-state transforms for arbitrary trace-class initial states, without that spectral hypothesis. On a common analytic domain, a perturbative force-fit equation has a first-order quotient pole at a simple baseline state-transform zero zeta only when the first-order inhomogeneous numerator I_tilde^(1)(zeta) is nonzero; at finite perturbation the actual zero and numerator must be checked, while matrix reconstructions additionally require adjugate non-cancellation. For rational reconstructions, the printed local or global clearing and no-cancellation hypotheses convert a rational kernel pole into a genuine upper-half-plane propagator pole. Such a pole is incompatible with a uniformly bounded CPTP family when the transforms agree on an open set, with Vieta's formula giving the imaginary-part budget. We formalize these paper-specific implications in Lean 4 from named classical inputs. A finite Jaynes--Cummings calculation provides a shifted, distributional KK check with a 0.024 percent fixed-truncation residual.