Topological methods in solvability theory of multidimensional pair integral operators with homogeneous kernels of compact type

The problem of getting effective Fredholm conditions for operators with bihomogeneous kernels reduces to the question of invertibility for families of operators with homogeneous kernels and to the calculation of homotopy invariants for spaces of Fredholm and invertible operators of that type. The purpose of the present paper is to study integral operators with homogeneous kernels of compact type in $L_p(\mathbb R^n)$, $1<p<+\infty$. The classes of homotopy equivalence for the spaces of Fredholm and invertible operators in the $C^*$-algebra of pair operators with homogeneous kernels of compact type are calculated by means of operator $K$-theory.