  The goal of this paper is to study the structure of noncommutative weighted
shifts, their properties, and to understand their role as models (up to
similarity) for $n$-tuples of operators on Hilbert spaces as well as their
implications to function theory on noncommutative (resp.commutative) Reinhardt
domains. We obtain a Rota type similarity result concerning the joint
similarity of $n$-tuples of operators to parts of noncommutative weighted
multi-shifts and provide a noncommutative multivariable analogue of
Foias-Pearcy model for quasinilpotent operators. The model noncommutative
weighted multi-shift which is studied in this paper is the $n$-tuple
$W=(W_1,\ldots, W_n)$, where $W_i$ are weighted left creation operators of the
full Fock space with $n$ generators associated with a weight sequence
$\boldsymbol \mu=\{\mu_\beta\}_{|\beta|\geq 1}$ of nonnegative numbers.
  We also represent the injective weighted multi-shifts $W_1,\ldots, W_n$ as
ordinary multiplications by $Z_1,\ldots, Z_n$ on a Hilbert space of
noncommutative formal power series. This leads naturally to analytic function
theory in several complex variables. One of the goal for the remainder of the
paper is to analyze the extent to which our noncommutative formal power series
represent analytic functions in several noncommutative (resp. commutative)
variables and to develop a functional calculus for arbitrary $n$-tuples of
operators on a Hilbert space.
