Hypercontractions
Hypercontractions are a class of linear operators on a Hilbert space that generalize the notion of contractions by satisfying a family of stronger positivity conditions. The basic contraction condition is the inequality I − T* T ≥ 0, which defines a contraction. Hypercontractions of order m require the operator polynomial P_m(T) = ∑_{j=0}^m (−1)^j binom(m, j) T*^j T^j to be positive semidefinite. This creates a hierarchy of increasingly restrictive conditions, with contractions corresponding to the first order in the sequence.
These conditions have implications for dilation and function theory. Many hypercontractions admit dilations to larger spaces
In practice, hypercontractions provide a framework for analyzing stable systems and for classifying operator tuples according
See also: contraction, dilation theory, operator theory, reproducing kernel Hilbert spaces.