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I didn't know about the need to be pseudolocal to work out a classical index theory. Do you have any reference (book&chapter or article) about the topic? And also, would you agree with me that these operators (at least the one in the example) don't have any principal symbol?
You are right. By "Hörmander symbol" I mean that the symbol is "in the class $S_{1,0}^m$ of Hörmander", using the wording of Wikipedia. I hope there is no big ambiguity in this definition.