On an algebraic method in the study of integral equations with shift

Vasile Neaga

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2006), pp. 69-74

Résumé

The work is centred on the sdudy of algebra $\mathfrak{A}$ generated by singular integral operators with shifts with continuous coefficients. We determine the set of maximal ideals of quotient algebra $\hat{\mathfrak A}$, $\hat{\mathfrak A}=\mathfrak{A}/\mathfrak{T}$, with respect to the ideal of compact operators. Prove that the bicompact of maximal ideals of $\hat{\mathfrak A}$ is isomorphic to the topological product $(\Gamma\times j)\times(\Gamma\times k)$, where $j=\pm 1$ and $k=\pm 1$. Necessary and sufficient condition are established for operators of $\mathfrak{A}$ to be noetherian and to admit equivalent regularization in space $L_p(\Gamma,\rho),$ regularizators for noetherian operators are constructed. The study is done in the space $L_{p}(\Gamma,\rho)$ with weight $\rho(t)=\prod\limits_{k=1}^{n}|t-t_{k}|^{\beta^{k}}$ and is based on the theory of Ghelfand [1] concerning Banach algebras.