Riesz bases of eigenfunctions and adjoint eigenfunctions of some integral operators (Q1874930)

Consider the integral operator \[ A f(x) = \int_0^{1-x} ( 1+ A(1-x, t)) f(t) dt , \quad x \in [0, t], \tag{1} \] acting in \(L_2 [0, 1]\). The author shows that the root vectors of the operator \(A\) form a Riesz basis in \(L_2 [0, 1]\) under the assumptions that the function \(A(x, t)\) and its first order derivative with respect to \(x\) are continuous for \(t\), \(0 \leq t \leq x\), \(\partial^2 A/ \partial x \partial t\) is uniformly bounded for all \(t\) and for almost all \(x\), and the condition \[ \frac{\partial^k A(x, t)}{\partial x^k}\biggl |_{t=x} =0, \quad k=0, 1 \tag{2} \] is satisfied. Similar integral operators with kernels like Green's functions, i.e., kernels with a discontinuous \((n-1)\)th-order derivative with respect to \(x\) on the diagonal \(t=x\), were considered by \textit{A. P. Khromov} [Mat. Sb., Nov. Ser. 114(156), 378-450 (1981; Zbl 0488.45016)] and operators (1), (2) are studied by \textit{A. P. Gurevich} [Voron. Zimnyaya Mat. Shkola. Tez. Dokl. (Abstr. Voronezh Winter Math. School) (Voronezh) (1995)]. Attention in these works was mostly focused on inversion of integral operators and the equiconvergence of spectral expansions in eigenfunctions and adjoint eigenfunctions, and in ordinary trigonometric series.