Is There a Spectral Theory for All Bounded Linear Operators?

Abstract

What Is Spectral Theory? By spectral theory we mean the theory of structure of certain bounded linear operators on a Hilbert space. In a broader sense, the history of spectral theory goes way back to the nineteenth century, when the objects of study used to be infinite systems of linear equations and integral equations. The subject was revolutionized in the late 1920s by von Neumann, when he defined the notion of an abstract Hilbert space and considered bounded linear operators on it. In this modern sense a successful spectral theory was soon obtained by Riesz for all compact operators as a direct extension of the theory of finite square matrices. By the early 1930s, von Neumann had obtained a satisfactory spectral theory for all normal operators, with self adjoint (or Hermitian) operators and unitary operators as important special cases. Spectral theory has evolved further since then; a vast amount of work was done on extending spectral theory to various Banach algebras. However, efforts to extend spectral theory to all bounded linear operators on a Hilbert space have met with resistance so far. This will be the problem we concentrate on in this article. To be explicit, the problem we consider is the existence of a