Sz.-Nagy Foias theory: Dilation of contractions on a Hilbert space, minimal
isometric dilation, unitary dilation. Von Neumann’s inequality.

Ando’s theorem: simultaneous dilation of a pair of commuting contractions. Parrott’s example
of a triple of contractions which cannot be dilated simultaneously. Creation
operators on the full Fock space and the symmetric Fock space.

Operators spaces. Completely positive and completely bounded maps.
Endomorphisms. Towards dilation of completely positive maps. Unbounded
operators: Basic theory of unbounded self-adjoint operators.

Suggested books and references:

John B. Conway, A course in Functional Analysis
,Springer, 1985.

Vern Paulson, Completely Bounded Maps and Dilations
,Pitman Research Notes, 1986.