Certain indicators of the quasi-nilpotency of functional operators (Q1589165)

This article presents some modifications of the theorem about the vanishing of the spectral radius of the linear Volterra operator \(G\) in \(L_\infty\) in terms of the existence of increasing chains \(\{H_0, H_1,\dots, H_k\}\) of invariant subspaces such that \(\|P_{H_i/H_{i-1}}\|< \delta\), \(i= 1,\dots, k\). These modifications are applied to the study of the linear operator with partial integrals; this operator arises the Riemann problem for the equation \[ x_{t^1t^2}= f(t,x(t), x_{t^1}(t), x_{t^2}(t))\qquad (t\in \mathbb{R}^2) \] in a special domain.