# Journal of Operator Theory

Volume 37, Issue 2, Spring 1997 pp. 263-279.

On two-dimensional singular integral operators with conformal Carleman shift**Authors**: R. Duduchava (1), A. Saginashvili (2) and E. Shargorodsky (3)

**Author institution:**(1) A. Razmadze Mathematical Institute, Academy of Sciences of Georgia, 1, M. Aleksidze str., Tbilisi 93, GEORGIA

(2) A. Razmadze Mathematical Institute, Academy of Sciences of Georgia, 1, M. Aleksidze str., Tbilisi 93, GEORGIA

(3) A. Razmadze Mathematical Institute, Academy of Sciences of Georgia, 1, M. Aleksidze str., Tbilisi 93, GEORGIA. Current address: Department of Mathematics, Kings College London, Strand, London, WC2R 2LS, U.K.

**Summary:**For the class of singular integral operators with continuous coefficients and with the conformal shift over a two-dimensional bounded domain $G \subset \mathbb C$ an explicit Fredholm property criterion is obtained. Operators under consideration have kernels $[(\bar \varsigma - \bar z)/(\varsigma - z)]^k \left| {\varsigma - z} \right|^{ - 2}$ either with positive or with negative $k \in \mathbb Z\backslash \{0\}$; the conformal shift $W\varphi (z) = \varphi (\omega (z))$, $\omega : G \to G$ is of Carleman type: $W^k \ne I$ for k = 1, 2, ..., n – 1 and W^n = I. It is proved also that a Fredholm operator A of such type has trivial index Ind A = 0.

**Keywords:**Singular integral operators, two-dimensional domains, Carleman shift.

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