Unbounded normal operators in Hilbert space

Abstract

The theory of abstract unbounded operators and their extensions developed along two main lines. The first is the theory of extensions of symmetric operators (Achiezer and Glasman [3]). In this case the problem is to extend a symmetric operator to a self-adjoint one. This theory includes necessary and sufficient conditions for the existence of such extensions, as well as complete characterization of the nature of these extensions, especially the nature of the spectrum of various extensions pertaining to the same operator. This theory imposes severe conditions and can be applied only in particular cases. In order to deal with abstract problems arising in the theory of partial differential equations and integral equations a relaxation is needed. The notions of generalized solution, solution, etc. (for example in Lax [1]) and the theory of distributions lead to the following problem of extension of general abstract operators. (For the sake of simplicity the illustration below will be confined to operators acting in Hilbert space.) The operator (differential, integral, etc.) !T is embedded in a certain space E. In this space a domain Do is considered. Denote Y acting on Do by To. To is in general bounded below on Do so it may be closed to an operator To ' To acting on Do D bo. By this process the generalized are obtained. In most cases the range of To is not the whole space so an extension is needed. Among all possible extensions only the are considered. They are obtained by looking at a natural formal adjoint 9* defined on the closure of the set bo. Denote this operator by T* and denote (T1*)* by T1. Any function in the domain of T1 is called a solution. In general the range of T1 is the whole space but there are too many weak solutions because there are null solutions. So the problem is to find an intermediate To c T c T1 having a bounded inverse defined on the whole space. Somewhat different versions are in G. C. Rota [4] and Gohberg and Krein [5]. These general theories have only weak results, because of the generality of the operators considered.