We study spectral properties of 2 × 2 block operator matrices whose entries are unbounded operators between Banach spaces and with domains consisting of vectors satisfying certain relations between their components. We investigate closability in the product space, essential spectra and generation of holomorphic semigroups. Application is given to several models governed by ordinary and partial differential equations, for example containing delays, floating singularities or eigenvalue dependent boundary conditions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)