On Boundary Value Problems for Dirac Type Operators: I. Regularity and Self-Adjointness

In a series of papers, we will develop systematically the basic spectral theory of (self-adjoint) boundary value problems for operators of Dirac type. We begin in this paper with the characterization of (self-adjoint) boundary conditions with optimal regularity, for which we will derive the heat asymptotics and index theorems in subsequent publications. Along with a number of new results, we extend and simplify the proofs of many known theorems. Our point of departure is the simple structure which is displayed by Dirac type operators near the boundary. Thus our proofs are given in an abstract functional analytic setting, generalizing considerably the framework of compact manifolds with boundary. The results of this paper have been announced previously by the authors (J. Brüning and M. Lesch, in “Geometric Aspects of Partial Differential Equations (B. Booss- Bavnbek and K. P. Wojciechowski, Eds.), Contemporary Mathematics, Vol. 242, pp. 203–205, Amer. Math. Soc., Providence, RI, 1999).