On Choosing a Physically Meaningful Topological Classification for Non-Hermitian Systems and the Issue of Diagonalizability

The topological classification of hermitian operators is solely determined by the presence or absence of certain discrete symmetries. For non-hermitian operators we in addition need to specify the type of spectral gap. They come in the flavor of a point gap or a line gap. Since the presence of a line gap implies the existence of a point gap, there is usually more than one mathematical classification applicable to a physical system. That raises the question: which of these gap-type classifications is physically meaningful? To decide this question, I propose a simple criterion, namely the choice of physically relevant states. This generalizes the notion of Fermi projection that plays a crucial role in the topological classification of fermionic condensed matter systems, and enters as an auxiliary quantity in the bulk classification of photonic and magnonic crystals. After that the classification is entirely algorithmic, the system's topology is encoded in (pairs of) projections with symmetries and constraints. A crucial point in my investigation is the relevance of diagonalizability. Even for existing topological classifications of non-hermitian systems diagonalizability needs to be assumed to ensure that continuous deformations of the hamiltonian lead to continuous deformations of the spectra, projections and unitaries.