OAI/Analysis/KadisonKastler/PublishedHardyEvaluation.lean

Published evaluation of causal Fourier multipliers

Kalle M. Mikkola, Fourier multipliers for L² functions with values in nonseparable Hilbert spaces and operator-valued Hᵖ boundary functions, Ann. Acad. Sci. Fenn. Math. 33 (2008), 121–130, Proposition 1.3 and Theorem 1.5, pp.123–124. The Poisson representation is the Hardy-space representation theorem cited in the proof of Corollary 1.7, p.125: Rosenblum–Rovnyak, Hardy Classes and Operator Theory (1985), Theorem 4.8B, p.90.

We retain only evaluation at one point, on the concrete algebra of positive-frequency-preserving multiplication operators. Conjugation by the Fourier transform identifies it with the causal translation-invariant operators of Proposition 1.3 (a reflection interchanges Fourier signs). That proposition gives the algebra representation; Theorem 1.5 identifies the boundary field, and Poisson representation identifies its evaluation. No holomorphic function space, norm estimate, perturbation conclusion or choice of analytic approximant is included in this input.