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Index formula for convolution type operators with data functions in \(\operatorname{alg}(SO,PC)\). (English) Zbl 1358.47009

Summary: We establish an index formula for the Fredholm convolution type operators \(A = \sum_{k = 1}^m a_k W^0(b_k)\) acting on the space \(L^2(\mathbb{R})\), where \(a_k\), \(b_k\) belong to the \(C^\ast\)-algebra \(\operatorname{alg}(SO,PC)\) of piecewise continuous functions on \(\mathbb{R}\) that admit finite sets of discontinuities and slowly oscillate at \(\pm\) first in the case where all \(a_k\) or all \(b_k\) are continuous on \(\mathbb{R}\) and slowly oscillating at \(\pm\) and then assuming that \(a_k, b_k \in \operatorname{alg}(SO,PC)\) satisfy an extra Fredholm type condition. The study is based on a number of reductions to operators of the same form with smaller classes of data functions \(a_k\), \(b_k\), which include applying a technique of separation of discontinuities and eventually lead to the so-called truncated operators \(A_r = \sum_{k = 1}^m a_{k, r} W^0(b_{k, r})\) for sufficiently large \(r > 0\), where the functions \(a_{k, r}, b_{k, r} \in PC\) are obtained from \(a_k, b_k \in \operatorname{alg}(SO,PC)\) by extending their values at \(\pm r\) to all \(\pm t \geq r\), respectively. We prove that \(\operatorname{ind} A = \lim_{r \to \infty} \operatorname{ind} A_r\) although \(A = \text{s}-\lim_{r \to \infty} A_r\) only.

MSC:

47A53 (Semi-) Fredholm operators; index theories
46H05 General theory of topological algebras
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