19.8.1 Integral equations

Problem: The integral operator \( (A \varphi)(x)=\int_{0}^{1} \varphi(y) d y \) is considered as acting from \( L_{2}[0 ; 1] \) in \( L_{2}[0 ; 1] \). Is the operator \( A \) completely continuous? Is the operator \( A \) self-adjoint? Find all characteristic values of the operator \( A \) and the corresponding eigenfunctions. For what functions \( f \in L_{2}[0 ; 1] \) is the equation \( (A \varphi)(x)=f(x), 0 \leq x \leq 1 \) solvable, with respect to \( \varphi \in L_{2}[0 ; 1] \) ?