2004/12/08 by Dmitry S. Kalyuzhny\uı-Verbovetzki\uı, Kalyuzhny\uı-Verbovetzki\uı, Dmitry S.
Mathematics · #47A13 #47A20 #47A56 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:47A13 #msc:47A20 #msc:47A56
paper · pdf · doi:10.48550/arxiv.math/0412163
arxiv created 2004/12/08 · arxiv updated 2009/12/01
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 \widetildeA=(\widetildeA1,...,\widetildeAN) for which ζ\widetildeA:=ζ1\widetildeA1+... +ζN\widetildeAN is a unitary operator for each ζ:=(ζ1,...,ζN) in the unit torus \mathbbTN. 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 \mathbbDN, 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)∈ Cρ,N which is not simultaneously similar to any T=(T1,...,TN)∈ C1,N, however if A∈ Cρ,N admits a uniform unitary ρ-dilation then A is simultaneously similar to some T∈ C1,N.