Abstract
We suggest a new version of the notion of ρ-dilation (ρ > 0) of an N-tuple A = (A1, . . . , AN) of bounded linear operators on a common Hilbert space. We say that A belongs to the class Cρ,N if A admits a ρ-dilation Ã= (Ã1, . . . ,Ã;N) for which ζÃ:= ζ1ζ1 + ⋯ + ζNÃN is a unitary operator for each ζ := (ζ1, . . . , ζN) in the unit torus TN. For N = 1 this class coincides with the class Cρ of B. Sz.-Nagy and C. Foiaş. We generalize the known descriptions of Cρ,1=Cρ to the case of Cρ,N, N > 1, using so-called Agler kernels. Also, the notion of operator radii wρ, 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, . . . , AN) ε Cρ,N which is not simultaneously similar to any T = (T1, . . . , TN) ε C1,N, however if A ε Cρ,N admits a uniform unitary ρ-dilation then A is simultaneously similar to some T ε C1,N.
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Dedicated to Israel Gohberg on his 75th birthday
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Kalyuzhnyi-Verbovetzkii, D.S. (2005). Multivariable ρ-contractions. In: Gohberg, I., et al. Recent Advances in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 160. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7398-9_13
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DOI: https://doi.org/10.1007/3-7643-7398-9_13
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