Special Classes of Linear Operators

Abstract

The first part introduces special classes of bounded linear operator while the second part presents some results on Hamilton operators of quantum physics. Orthogonal projections or projectors are self-adjoint bounded linear operators P which are idempotent. They are characterized by their range. Isometric operators are bounded linear operators from one Hilbert space to another which do not change the length of vectors. If an isometric operator is surjective it is a unitary operator. A unitary operator $U:\mathcal{H}{\to} \mathcal{K}$ is characterized by the relations $U^*U =id_{\mathcal{H}}$ and $UU^*=id_{\mathcal{K}}$. Next one parameter groups of unitary operators are introduced and under a mild continuity assumption the self-adjoint generator of such a group is determined in Stone’s theorem. Another important application of unitary operators is the mean ergodic theorem of von Neumann. We present it together with the early results of ergodic theory: Poincaré recurrence theorem and Birkhoff’s strong ergodic theory. Given a free self-adjoint Hamilton operator $H_0=\frac{1}{2m} p^2$ one often needs to know under which conditions on the potential the Hamilton $H=H_0+ V(q)$ is self-adjoint in $L^2(\mathbb{R}^3)$. Sufficient condition are given in the Kato-Rellich theorem using the concept of a Kato perturbation. It follows that H is self-adjoint whenever V is a real-valued function in $L^2(\mathbb{R}^3) + L^{\infty}(\mathbb{R}^3)$.