Joint similarity in admissible noncommutative polydomains

Abstract

In this paper we solve several problems concerning joint similarity to tuples of bounded linear operators in noncommutative polydomains Dg−1(H)⊂B(H)n1+⋯+nk associated with admissible k-tuples g:=(g1,…,gk) of formal power series gi:=1+∑αi∈Fni+,|αi|≥1bi,αZi,α, biα>0, in noncommuting indeterminates Zi,1,…,Zi,ni, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. We obtain analogues of the Rota's model theorem for operators with spectral radius less than one, the refinement obtained by Foiaş and by de Branges and Rovnyak for similarity to strongly stable contractions, and the Sz.-Nagy characterization of operators similar to isometries (or unitary operators). The results are new even in the particular case when k=1 for certain classes of admissible noncommutative domains which were recently introduced. Joint similarity problems to tuples of operators in noncommutative varieties in admissible polydomains are also discussed. This includes, in particular, the commutative setting.