We will consider in this talk non-self-adjoint operators in Hilbert spaces given as relatively compact perturbations of a self-adjoint operator. Typical examples are Schrödinger operators with bounded, complex potentials vanishing at infinity. We will describe abstract conditions ensuring that the Hilbert space admits a direct sum decomposition into H-invariant subspaces, generalizing the well-known spectral decomposition of self-adjoint operators in terms of their spectral measures. A central role in the talk will be played by spectral singularities, an abstract notion corresponding to that of real resonances for Schrödinger operators. We will also present a useful regularized functional calculus for non-self-adjoint operators. This is joint work with Nicolas Frantz.

Meeting ID: 814 591 7621 Passcode: 147983