Linear Operators

Abstract

In our approach to linear operators we stress the fact that for the specification of a linear operator A it is important to specify first a linear subspace as its domain of definition D(A) accurately and then how points of this domain are assigned to points in the target space. Thus, the same assignment on different subspaces as domains defines different linear operators. This is illustrated by several examples. Section 20.1 defines closed and closable operators and we prove the existence of a unique adjoint $A^*$ for a densely defined linear operator A. And it is shown that densely defined linear operators whose adjoints are also densely defined are closable and that the closure is given by its bi-adjoint $A^{**}$. Section 20.3 defines symmetric, self-adjoint, and essentially self-adjoint linear operators and studies their relationships. Furthermore, important criteria for self-adjointness of a symmetric operator are proven. Section 20.4 illustrates these concepts by some examples: multiplication operators, momentum operator, and the free Hamilton operator.