Unitary operators on Hilbert spaces are generally regarded as well-understood objects in analysis. A fundamental theorem of Sz.-Nagy says that every contraction \(T\) on a Hilbert space \(H\) can be viewed as part of a unitary operator \(U\) acting on a larger Hilbert space \(K\) containing \(H\). This idea is known as the unitary dilation of a contraction.
In this talk, we discuss the basic philosophy of dilation theory and explain how it connects to some important questions in analysis. As a main application, we discuss von Neumann’s inequality, which states that for every polynomial \(p\) and every contraction \(T\), the norm of \(p(T)\) is bounded by the supremum norm of \(p\) on the unit circle. Time permitting, we will also briefly mention a few further consequences and applications of dilation theory.
