Journal Article

C-ALGEBRAS OF POLY-BERGMAN TYPE OPERATORS WITH PIECEWISE SLOWLY OSCILLATING COEFFICIENTS

Karlovich, Yuri I, Espinoza-Loyola, Enrique

2022

Overview

Given n,m∈N and a simply connected uniform domain U⊂C with a sufficiently smooth boundary ∂U , we study the C* -algebra BU,n,m(L):=algaI,BU,1,…,BU,n,B~U,1,…,B~U,m:a∈X(L), generated by the operators of multiplication by functions in X(L) , by the poly-Bergman projections BU,1,…,BU,n and by the anti-poly-Bergman projections B~U,1,…,B~U,m acting on the Lebesgue space L2(U) . The C* -algebra X(L) is generated by the set SO∂(U) of all bounded continuous functions on U that slowly oscillate at points of ∂U and by the set PC(L) of all piecewise continuous functions on the closure U¯ of U with discontinuities on a finite union L of piecewise Dini-smooth curves that have one-sided tangents at every point z∈L , possess a finite set Y=L∩∂U , do not form cusps, and are not tangent to ∂U at the points z∈Y . Making use of the Allan-Douglas local principle, the limit operators techniques, quasicontinuous maps, and properties of SO∂(U) functions, a Fredholm symbol calculus for the C* -algebra BU,n,m(L) is constructed and a Fredholm criterion for its operators is obtained.

• Publisher: Springer Nature B.V
• Subject: Algebra / Continuity (mathematics) / Mathematical analysis / Multiplication / Operators (mathematics) / Smooth boundaries / Tangents