Operator Domains in Nonclassical Measurement Error: A Comment on Hu and Schennach (2008)

Abstract

Hu and Schennach (2008) identify a model with nonclassical measurement error through similarities of observable integral operators and latent multiplication operators. Their proof treats the similarities and spectral projections as bounded on the ambient L1 space. A wrapped Gaussian model satisfies Assumptions 1–5 while the observable similarity and displayed projections are unbounded; the density columns also lie outside the inverse operator's domain. Conjugated latent-band projections extend to a bounded spectral measure exactly under a uniform band-stability inequality. Combined with outcome separation and dense range, full band stability makes the measurement operator bounded below, which conflicts with smoothing by bounded densities on a nonatomic support. Equipping each candidate range with its transported graph norm restores the operator algebra. Reconstruction can then proceed under common-resolution or cyclic-orbit conditions. The factorization conclusion under Assumptions 1–5 remains an identification question.

Technical point

A model satisfying Assumptions 1–5 can still produce an unbounded observable similarity, unbounded conjugated spectral projections, and density columns outside the inverse operator's domain.

Scope