Dilation theory studies a badly behaved operator by realising it as the compression of a better behaved one acting on a larger space: T = P_H U|_H, where U is unitary (or normal, or isometric) on a Hilbert space K containing H, and the compression reproduces every power of T. Sz.-Nagy's theorem, that every Hilbert-space contraction has a unitary power dilation, is the founding result; it yields von Neumann's inequality, the Sz.-Nagy-Foias functional model, and commutant lifting. The same paradigm extends far beyond single contractions: to commuting tuples (Ando's theorem in two variables and its failure in three), to completely positive maps via Stinespring's theorem and Arveson's extension theorem, to one-parameter and more general semigroups of contractions and of CP maps (E-semigroups and E_0-dilations), to representations of operator algebras through the C*-envelope and boundary representations, and to noncommutative and matrix-convex settings where dilation questions become questions about matrix convex sets, joint measurability of quantum observables, and semidefinite feasibility. The unifying pedagogical point is that a dilation converts an intractable question about a general operator into a computation with a normal or unitary object whose spectral theory is fully understood.
A 70-page expository survey of dilation theory in operator theory and operator algebras, written for the IWOTA 2019 proceedings. A dilation of an operator T acting on a Hilbert space H is an operator…