Closed and Adjoint Operators

Abstract

Chapter 1 deals with closed operators, closable operators, and adjoint operators. Closed operators and closable operators are large classes of operators that cover almost all interesting unbounded operators occurring in applications. The Hilbert space scalar product allows one to define the adjoint of a densely defined linear operator. Various characterizations of closed and closable operators and basic results on adjoint operators are derived. These notions are discussed in great detail for the differentiation operators $-\mathrm {i}\frac{d}{dx}$ and $-\frac{d^{2}}{dx^{2}}$ on (bounded and unbounded) intervals and for linear partial differential operators with C ^∞-coefficients acting on domains of ℝ^d. The differentiation operators with various boundary conditions are guiding examples that occur repeatedly throughout the book. In the final section of Chap. 1, invariant subspaces and reducing subspaces of unbounded linear operators are defined and characterized.