On the algebra generated by Volterra integral operators with homogeneous kernels and continuous coefficients

O. G. Avsyankin ; G. A. Kamenskikh

Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 4, pp. 19-29

Résumé

We consider Volterra multidimensional integral operators with continuous coefficients in Lebesgue spaces. It is assumed that the kernel of the integral operator is homogeneous of degree $(-n)$, invariant under the rotation group $SO(n)$ and satisfies a certain summability condition that ensures the boundedness of the operator. In this paper, the main object of research is the Banach algebra $\mathfrak{A}$ generated by all operators of the above type and the identity operator. The algebra $\mathfrak{A}$ is noncommutative, and for its study we turn to the quotient algebra $\mathfrak{A}/\mathfrak{T}$, where $\mathfrak{T}$ is the set of all compact operators. It is shown that the algebra $\mathfrak{A}/\mathfrak{T}$ is commutative, which makes it possible to apply the general methods for studying commutative Banach algebras. In particular, a description of the maximal ideals space of the algebra $\mathfrak{A}/\mathfrak{T}$ is given and a criterion for the invertibility of elements from this algebra is found. Based on this, we construct a symbolic calculus for the Banach algebra $\mathfrak{A}$ that is, each operator from this algebra is assigned a certain continuous function. This function is called the symbol of the operator. In terms of the symbol, we obtained necessary and sufficient conditions for the Fredholm property of an operator from $\mathfrak{A}$, as well as an index formula.