On the algebra, generated by an abstract singular operator and a Carleman shift. (Russian. English summary) Zbl 0566.47020
The algebra \({\mathcal R}\) generated by an abstract singular operator S, Carleman shift and continuous coefficients is investigated. For the algebra \({\mathcal R}\) the symbol algebra Sym \({\mathcal R}\) has been constructed. The Noetherian criterion for operators from \({\mathcal R}\) is given in symbol terms. In contrast to a well-known case (S is a one- dimensional singular integral operator) the symbol algebra Sym \({\mathcal R}\) here is a \(C^*\)-algebra with irreducible representations of one-, two- or four-dimensions. Some concrete examples have been considered.
MSC:
47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
47A53 | (Semi-) Fredholm operators; index theories |
47Gxx | Integral, integro-differential, and pseudodifferential operators |
45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
47C15 | Linear operators in \(C^*\)- or von Neumann algebras |
46H05 | General theory of topological algebras |