Abstract:
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.
Keywords and phrases:Banach algebras, noetherian singular operators, regularization of operator.
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