C*-algebras of Bergman-type operators with piecewise continuous coefficients and shifts

摘要

Let U be a bounded simply connected domain in C with sufficiently smooth boundary Γ. Let G be a commutative group of conformal mappings of ū onto itself which is similar to the group of elliptic, hyperbolic or parabolic mappings of the closed unit disc D? onto itself, and let L _G be a G-invariant set of simple Lyapunov curves in ū such that for every z ∈ ū an at most finite number of curves in L _G are intersecting at z and at every z ∈ Γ these curves form with Γ pairwise distinct angles lying in (0, π). Let A _(n,m) (U, L _G) be the C*-algebra generated by n poly-Bergman projections, m anti-poly-Bergman projections and by all multiplication operators aI acting on the space L ~2(U), where a are piecewise continuous functions on ū with possible discontinuities on subsets of L _G that are continuous at common fixed points of g ∈ G. For mentioned groups G, applying a local-trajectory method and a Fredholm symbol calculus for the C*-algebra A n,m (U, L _G), we establish Fredholm criteria for the operators B in the C*-algebras B generated by all operators A ∈ A _(n,m)(U, L _G) and all weighted shift operators W _g(g ∈ G), where W _gf = g′(f ○ g) for f ∈ L ~2(U).