On the Algebra Generated by Volterra Integral Operators with Homogeneous Kernels and Continuous Coefficients

Abstract

We consider Volterra multidimensional integral operators with continuous coefficients in Lebesgue spaces. The kernel of an integral operator is assumed homogeneous of degree $(-n)$, invariant under the rotation group $SO(n)$, and satisfying some summability condition that ensures the boundedness of the operator. The main object of research is the noncommutative Banach algebra $\mathfrak{A}$ generated by all operators of the above type and the identity operator. To study $\mathfrak{A}$, we turn to the quotient algebra $\mathfrak{A}/\mathfrak{T}$, with $\mathfrak{T}$ the set of all compact operators. We prove that $\mathfrak{A}/\mathfrak{T}$ is commutative, which allows us to apply the general methods of commutative Banach algebras. In particular, we describe the space of maximal ideals of $\mathfrak{A}/\mathfrak{T}$ and find some invertibility criterion for the elements of $\mathfrak{A}/\mathfrak{T}$. We then construct some symbolic calculus for $\mathfrak{A}$ on assigning to each operator in $\mathfrak{A}$ a continuous function, the symbol of the operator. Using the symbols, we obtain the necessary and sufficient conditions for the Noetherian property of an operator in $\mathfrak{A}$, as well as an index formula.