Operators and Their Graphs

Abstract

Applications of operator theory in other branches of mathematics and in mathematical physics very often involve operators which are not bounded. This poses numerous difficulties whose source is, for the most part, the lack of useful algebraic structure on the set of unbounded operators. Our presentation of the theory of unbounded operators on a Hilbert space will focus on a few select issues and our preferred strategy for dealing with them will be to reduce them to questions about bounded operators. We will begin with some introductory information gathered in Sect. 8.1. In the following chapters we will introduce our key tool which we call the z-transform and later use this tool to extend various versions of the spectral theorem to unbounded self-adjoint operators. The final chapters will be devoted to several classical topics like self-adjoint extensions of symmetric operators and elements of the theory of one-parameter groups of unitary operators.