The motivating point of our research was differential operators on the noncommutative torus studied by Connes [1, 2], who in particular obtained an index formula for such operators. These operators include shifts (more precisely, in this case, irrational rotations); hence our interest in general differential equations with shifts naturally arose. Let M be a smooth closed manifold. We consider o...

We use elementary algebraic properties of left, right multiplication operators to prove some deep structural left $m$-invertible, $m$-isometric, $m$-selfadjoint and other related classes Banach space operators, often adding value extant results.

In this note we establish the positivity of Green’s functions for a class of elliptic differential operators on closed, Riemannian manifolds

We prove that the index formula for b-elliptic cone differential operators given by Lesch in [6] holds verbatim for operators whose coefficients are not necessarily independent of the normal variable near the boundary. We also show that, for index purposes, the operators can always be considered on weighted Sobolev spaces.

For differential operators which are invariant under the action of an abelian group Bloch theory is the tool of choice to analyze spectral properties. By shedding some new non-commutative light on this we motivate the introduction of a noncommutative Bloch theory for elliptic operators on Hilbert C-modules. It relates properties of C-algebras to spectral properties of module operators such as b...

A theory of non-unitary unbounded similarity transformation operators is developed. To this end the class of J-unitary operators U is introduced. These operators are similar to unitary operators in their algebraic aspects but differ in their topological properties. It is shown how J-unitary operators are related to so-called J-biorthonormal systems and J-selfadjoint projections. Families {Uα} o...

The theory of the classical Jacobi forms on H × C has been studied extensively by Eichler and Zagier[?]. Ziegler[?] developed a more general approach of Jacobi forms of higher degree. In [?] and [?], Gritsenko and Krieg studied Jacobi forms on H × Cn and showed that these kinds of Jacobi forms naturally arise in the Jacobi Fourier expansions of all kinds of automorphic forms in several variable...