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. Other functional-calculus bounds on the disk that we consider, together with their connection to dilation theory, include those of Berger–Stampfli, Okubo–Ando and Drury. In connection with the Okubo–Ando estimate, we then discuss Paulsen's similarity theorem, which gives a natural framework for understanding similarity to contractive operators.

Prerequisites

Students are expected to be familiar with basic measure theory, functional analysis (Hilbert spaces, Hahn–Banach, Stone–Weierstrass, Gelfand–Naimark, ...) and complex analysis (Maximum modulus principle, 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).
• The lecture notes are regularly updated on CMS, so please make sure to check the platform frequently.
• Tutorials take place biweekly (Monday, 10-12, Zeichensaal U.39). The first tutorial is scheduled for May 4.
• There will be no lectures on May 14 (Christi Himmelfahrt) and June 4 (Fronleichnam). To compensate for these cancellations, the first tutorial on May 4 may be replaced by an additional lecture. Confirmation will follow.
• There are biweekly hand-in assignments. They are published online every second Friday at 12:00, and the deadline is two weeks later, also on Friday at 12:00. The first set will be available on CMS on Friday, April 17. You can submit assignments in teams of up to three people.
• To be eligible for the exam, you must obtain at least 50% of the total points from the hand-in assignments.

Literature

See here.