Linear Operators in Hilbert Spaces

Abstract

We recall some fundamental notions of the theory of linear operators in Hilbert spaces which are required for a rigorous formulation of the rules of Quantum Mechanics in the one-body case. In particular, we introduce and discuss the main properties of bounded and unbounded operators, adjoint operators, symmetric and self-adjoint operators, self-adjointness criterion and stability of self-adjointness under small perturbations, spectrum, isometric and unitary operators, spectral theorem, unitary group, decomposition of the spectrum, and Weyl’s theorem on the essential spectrum. The above topics are treated with the help of examples and exercises and avoiding complete generality. In some cases, e.g., for the spectral theorem, the results are formulated without proofs.