Operator Theory

Summer 2020, Grade: 1.0

Operator Theory (OT) is concerned with (bounded or unbounded) linear operators between (usually normed) topological vector spaces, whereas functional analysis deals with operators to the ground field i.e. with functionals. It has its roots at the beginning of the 20th century in the study of integral equations by Fredholm and subsequently in the mathematical development of quantum mechanics.

The inner machinery of the theory mostly deals with abstractly studying properties of operators that naturally come up in the context of integral equations (e.g. Volterra), in ODE & PDE (e.g. Sturm–Liouville theory, Laplacians), or quantum mechanics (e.g. momentum & energy operators). This is done by systematically weakening and strengthening the concepts of continuity.

A cornerstone of the subject is the use of so-called functional calculi: Similar to how the sum and multiplication of two operators have a very natural interpretation, other functional expressions such as f(A), where f is some function and A is a linear operator have meaning. The first example of this one encounters is probably the structure theorem for solutions of a homogeneous system of linear differential equations, where exp(tA) shows up.  The more well behaved the operator A, the worse the functions can behave which can be evaluated at A, and vice versa. A theory concerning which classes of functions can be evaluated at which types of operators is known as a functional calculus.

Another reason for their study is that functional calculi interact very nicely with spectra of linear operators, which are the infinite-dimensional analogous of eigenvalues of matrices. They are particularly important in the study of quantum mechanics, where observables are modelled by linear operators on a Hilbert space and their possible values by the eigenvalues of that operator.

Application: OT is the language of quantum mechanics and finds a lot of application there. On the other hand, it naturally shows up whenever function spaces are systematically studied, which happens in almost every sub-discipline.