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Joint similarity to operators in noncommutative varieties

In this paper we solve several problems concerning joint similarity to n-tuples of operators in noncommutative varieties in $[B(\cH)^n]_1$ associated with positive regular free holomorphic functions in $n$ noncommuting variables and with sets of noncommutative polynomials in $n$ indeterminates, where $B(\cH)$ is the algebra of all bounded linear operators on a Hilbert space $\cH$. In particular, if $f=X_1+...+X_n$ and $\cP=\{0\}$, the elements of the corresponding variety can be seen as noncommutative multivariable analogues of Agler's $m$-hypercontractions.