Some Classes of Unbounded Operators

Abstract

In Chap. 3, we introduce and begin the study of some fundamental classes of unbounded operators. The most important ones for this book are symmetric operators and self-adjoint operators. For a densely defined symmetric operator, the deficiency indices are defined and investigated, and the von Neumann formula about the domain of the adjoint is obtained. As a consequence, some self-adjointness criteria for symmetric operators are derived. Among others, we give a short proof of Naimark’s classical theorem, which states that each densely defined symmetric operator has a self-adjoint extension on a possibly larger Hilbert space. Further, we define classes of operators (sectorial operators, accretive operators, dissipative operators) that will appear later as operators associated with sectorial forms or as generators of contraction semigroups. The last section of this chapter contains a brief introduction to unbounded normal operators.