A note on spectral theory of integral-functional Volterra operators

The paper’s theorem and strategy are plausible, but the proof has gaps: (i) the contraction estimate reuses the constant q from |a(t)||α|^ε ≤ q|a(0)| to bound the sum of two operators, without actually controlling the Volterra integral part; the asserted bound “≤ q||v||_L” is not justified and even contains an α^n vs α^ε inconsistency; (ii) the operator A is claimed on C[−T,T] while K is only assumed on {s ≤ t}, which is insufficient to define ∫_0^t for t < 0. The candidate solution supplies a detailed fixed-point construction near 0 and a stepwise Volterra extension to [0,T], and it explicitly fixes the t < 0 domain issue by adding the needed assumption on K. Hence the model gives a correct and complete proof under natural hypotheses, whereas the paper’s proof is incomplete.