nhypercontraction

nhypercontraction, often written n-hypercontraction, is a notion in operator theory describing bounded linear operators that satisfy a finite positivity condition generalizing contractions and hypercontractions. Let T be a bounded operator on a complex Hilbert space H and let n be a positive integer. Define the n-th order defect operator Δ_T^{(n)} by Δ_T^{(n)} = sum_{k=0}^n (-1)^k binom(n,k) (T^*)^k T^k. The operator T is called an n-hypercontraction if Δ_T^{(n)} is positive semidefinite, i.e., Δ_T^{(n)} ≥ 0.

Special cases and examples: For n=1, Δ_T^{(1)} = I - T^*T, so T is a contraction iff Δ_T^{(1)} ≥ 0. For n=2, Δ_T^{(2)} = I - 2 T^*T + (T^*)^2 T^2, which is the classical hypercontraction condition. If T is an isometry (T^*T = I), then Δ_T^{(n)} = 0 for every n, so T is an n-hypercontraction for all n. The unilateral shift on the Hardy space H^2 is a canonical example of an isometry and hence, in particular, an n-hypercontraction for all n.

Connections and context: n-hypercontractions form a family studied in dilation theory and function theory. They are linked to positivity properties of operator-valued kernels and to models of operators on reproducing kernel Hilbert spaces related to the n-th power of classical kernels. In suitable settings, an n-hypercontraction admits a dilation to a shift-type operator on a larger Hilbert space, extending the classical dilation theory for contractions.

See also: Contraction (operator theory), Hypercontraction, Dilation theory, Reproducing kernel Hilbert space, Agler’s work on operator inequalities.