A technical comment on solvability claims for Sylvester-type operator equations

Abstract

This independent technical comment checks four groups of claims in A. Yakoub and A. Mennouni, “New results on the solvability of Sylvester-type operator equations,” Filomat 40 (2026), 371–396, DOI: 10.2298/FIL2602371Y. Bounded-operator counterexamples on the infinite-dimensional Hilbert space ℓ² disprove both parts of Theorem 2.1, the existence assertions in Corollaries 2.5 and 2.6, and the necessity directions in both parts of Theorem 2.3. We give exact conditions for the two proposed solutions in Theorem 2.3 and correct the sign in a block pseudoinverse printed in Example 5.1. These findings warrant an audit of later arguments that use the affected theorem; they do not by themselves disprove every later conclusion. The PDF and LaTeX source are included.