Dilation Theory: A Guided Tour

Dilation theory is a paradigm for studying operators by way of exhibiting an operator as compression another which in some sense well behaved. For example, every contraction can be dilated to (i.e., of) unitary operator, and on this simple fact penetrating non-normal has been developed. In the first part survey, I will leisurely review key classical results dilation single or several commuting operators, sample applications function theory. Then, second part, give rapid account plethora variants their applications. particular, discuss completely positive maps semigroups, algebraic approach last present relatively new problems noncommutative setting are related study matrix convex sets systems, motivated control These include dilating tuples noncommuting normal with specified joint spectrum. also describe recently studied problem determining optimal constant $$c = c_{\theta ,\theta '}$$ , such that pair unitaries U, V satisfying U e i? UV cU?, cV ?, where U?, ? satisfy commutation relation $$V'U' e^{i\theta '} U'V'$$ . The solution gives rise surprising application continuity spectrum almost Mathieu from mathematical physics.