Multivariable ρ-contractions

Abstract

We suggest a new version of the notion of ρ-dilation (ρ > 0) of an N-tuple A = (A₁, . . . , A_N) of bounded linear operators on a common Hilbert space. We say that A belongs to the class C_ρ,N if A admits a ρ-dilation Ã= (Ã₁, . . . ,Ã;_N) for which ζÃ:= ζ₁ζ₁ + ⋯ + ζ_NÃN is a unitary operator for each ζ := (ζ₁, . . . , ζ_N) in the unit torus T^N. For N = 1 this class coincides with the class C_ρ of B. Sz.-Nagy and C. Foiaş. We generalize the known descriptions of C_ρ,1=C_ρ to the case of C_ρ,N, N > 1, using so-called Agler kernels. Also, the notion of operator radii w_ρ, ρ > 0, is generalized to the case of N-tuples of operators, and to the case of bounded (in a certain strong sense) holomorphic operator-valued functions in the open unit polydisk $\mathbb{D}^N$, with preservation of all the most important their properties. Finally, we show that for each ρ > 1 and N > 1 there exists an A = (A₁, . . . , A_N) ε C_ρ,N which is not simultaneously similar to any T = (T₁, . . . , T_N) ε C_1,N, however if A ε C_ρ,N admits a uniform unitary ρ-dilation then A is simultaneously similar to some T ε C_1,N.