16/09/2016, 10:25 — 10:50 — Room 6.2.38, Faculty of Sciences of the Universidade de Lisboa
Catarina Carvalho, Universidade de Lisboa
Index formula for convolution type operators with piecewise continuous, slowly oscillating coefficients
We establish an index formula for Fredholm convolution type operators on $L^2(\mathbb{R})$ of the form $$A=\sum_{k=1}^m a_kW^0(b_k), \quad a_k,b_k\in \rm{alg}(SO,PC)$$ where $\rm{alg}(SO,PC)$ is the $C^*$-algebra of piecewise continuous functions on $\mathbb{R}$ that admit finite sets of discontinuities and slowly oscillate at $\pm\infty$.
First we consider the case where all $a_k$ or all $b_k$ are continuous on $\mathbb{R}$ and slowly oscillating at $\pm\infty$; then we assume that $a_k, b_k \in \rm{alg}(SO,PC)$ satisfy an extra Fredholm type condition to reduce to the above.
The study is based on a number of reductions to operators with smaller classes of coefficients, which include applying a technique of separation of discontinuities and eventually lead to the so-called truncated operators $A_r$, for sufficiently large $r\gt 0$, with $PC$ coefficients. We prove that $\operatorname{ind} A=\lim_{r\to\infty} \operatorname{ind} A_r$, which can be computed by classical results of Duduchava.
The talk is based on joint work with M. Amélia Bastos and Yuri I. Karlovich.
