Continuous and Compact Linear Operators

On the spectrum of the operator which is a composition of integration and substitution

“…To estimate the spectral radius r(V φ ) we recall (see [14]) some results on integral operators with nonnegative kernels. Let (Kf )(x) = …”

Discretization of regularized integral equations in L ∞

“…In the case of "very" discontinuous initial data (e.g., the kernel K(s,t) is only Σ-measurable in s, and/or y G L°°(D)) we are forced to look for a solution in the space 354 R. Lepp of bounded measurable functions (see, e.g., [15]). Then the natural question arises: how to understand the convergence of solutions of approximate problems to the solution of the initial problem, since the well-known projection methods do not work in L°°(D) (the space C(D) of continuous functions is not dense in L°°(D) and, thus, it is not clear how to construct a system of "nice" subspases {B n }, B\ C £ 2 C .…”

“…Consider the discretization of a regularized integral equation of the first kind in the case of a discontinuous kernel, so that we are forced to look for a solution χ in the space Ζ/°°(Ζ),Σ,σ) = L°°(D) of essentially bounded measurable functions [15]: find x(s), χ G L 00 (D), such that Since equation (1.1) is essentially ill-posed (the range space of the compact operator k,…”