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Published in:
2015 | OriginalPaper | Chapter
C ∗-Algebras of Two-Dimensional Singular Integral Operators with Shifts
Authors : Y. I. Karlovich, V. A. Mozel
Published in: Current Trends in Analysis and Its Applications
Publisher: Springer International Publishing
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Abstract
We construct a Fredholm symbol calculus for the C
∗-algebra \({\mathfrak{B}}\) generated by the C
∗-algebra \({\mathfrak{A}}\) of two-dimensional singular integral operators with continuous coefficients on a bounded closed simply connected domain \(\overline{U}\subset\mathbb{R}^{2}\) with Liapunov boundary and by all unitary shift operators W
g
where g runs through a discrete solvable group G=F⋊H of diffeomorphisms of \(\overline{U}\) onto itself, where F is a commutative group of conformal mappings, H={e,γ} and γ is similar to the shift \(z\mapsto \overline{z}\). As a result, we establish a Fredholm criterion for the operators \(B\in{\mathfrak{B}}\).