Norms of multiplication operators: answering Fialkow--Loebl question

What the math gives to ML

The paper gives an exact singular-value characterization of the norm of the two-sided operator S_{a,b}:x maps to axb. Its transferable asset is the coupling of the spectra of the left and right factors through the pointwise product of their singular values, rather than separate worst-case bounds. This suggests regularizing matrix-valued neural modules such as token mixers, adapters, covariance layers, and Kronecker-factored transformations using the spectrum of their combined amplification. The paper also shows that weak-Lp quasi-norms are not monotone under logarithmic submajorisation, cautioning against using weak-Schatten penalties as reliable stability measures.