Functional Calculus

Functional Calculus

Functional calculus addresses a simple but far-reaching question: given a linear operator A on a Banach or Hilbert space, what does it mean to apply a scalar function f to A, and how can properties of f be translated into operator-theoretic information about f(A)? 

We begin with a brief recap of the continuous functional calculus for normal operators on a Hilbert space, followed by a detailed treatment of its extension to the measurable functional calculus, where the full spectral theorem for normal operators is the underlying core result. 

We then introduce the holomorphic functional calculus for arbitrary elements of a Banach algebra. In the absence of adjoints (and normality), the construction relies on contour integration in Banach spaces. We prove Runge's theorem—a classical approximation result in complex analysis—and use it to establish uniqueness of the holomorphic functional calculus and, in particular, consistency with the continuous functional calculus.

The second part of the course focuses on dilation theory and functional-calculus bounds on the unit disk. We treat von Neumann's inequality as a guiding example, which can be obtained as a consequence of Sz.-Nagy's dilation theorem. We also briefly study the multivariable analogue of von Neumann's inequality, where the situation changes and leads to interesting phenomena. 

Other functional-calculus bounds on the disk that we consider, together with their connection to dilation theory, include those of Okubo–Ando and Drury. If time permits, we will consider functional-calculus bounds on other domains, leading to the notion of c-spectral sets.

Prerequisites

Students are expected to be familiar with basic measure theory, functional analysis (Hilbert spaces, Hahn–Banach, Gelfand–Naimark, ...) and complex analysis (Cauchy's integral theorem/formula, ...). Familiarity with the Riesz representation theorem for positive functionals on continuous function spaces is helpful, but the necessary details will be reviewed during the course.

Organizational information

The course is worth 4.5 ECTS and will run during the summer semester in 2026. The lectures will be held in English and take place once a week (Thursday, 14-16, Hörsal IV).