A note on numerical singular values of compositions with non-compact operators

Abstract

Linear non-compact operators are difficult to study because they do not exist in the finite-dimensional world. Recently, Hofmann and Mathé [Electron. Trans. Numer. Anal., 57 (2022), pp. 1–16] studied the singular values of the compact composition of the non-compact Hausdorff moment operator and the compact integral operator and found credible arguments, but no strict proof, that those singular values fall only slightly faster than those of the integral operator alone. However, the fact that numerically the singular values of the combined operator fall exponentially fast was not mentioned. In this note, we supply the missing numerical results and provide an explanation why the two seemingly contradictory results may both be true.