Essentially translation invariant pseudodifferential operators on manifolds with cylindrical ends

We study two classes (or calculi) of pseudodifferential operators defined on manifolds with cylindrical ends: the class of pseudodifferential operators that are ``translation invariant at infinity'' and the class of ``essentially translation invariant operators'' that have appeared in the study of layer potential operators on manifolds with straight cylindrical ends. Both classes are close to the $b$-calculus considered by Melrose and Schulze and to the $c$-calculus considered by Melrose and Mazzeo-Melrose. Our calculi, however, are different and, while some of their properties follow from those of the $b$- or $c$-calculi, many of their properties do not. In particular, the ``essentially translation invariant calculus'' is spectrally invariant, a property not enjoyed by the ``translation invariant at infinity'' calculus or the $b$-calculus. For our calculi, we provide easy, intuitive proofs of the usual properties: stability for products and adjoints, mapping and boundedness properties for operators acting between Sobolev spaces, regularity properties, existence of a quantization map and topological properties of our algebras, the Fredholm property. Since our applications will be to the Stokes operator, we systematically work in the setting of Agmon-Douglis-Nirenberg-elliptic operators.