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 $(\mathrm{\Gamma }\times j)\times (\mathrm{\Gamma }\times k)$, 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 $L_p(\mathrm{\Gamma },\rho ),$ regularizators for noetherian operators are constructed. The study is done in the space $L_p(\mathrm{\Gamma },\rho )$ with weight $\rho (t)=\prod _{k=1}^n|t-t_k{|}^{{\beta }^k}$ and is based on the theory of Ghelfand [1] concerning Banach algebras.

Bibliographie

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