  The noncommutative Choquet boundary and the C*-envelope of operator systems
of the form Span{1,T,T*}, where T is a Hilbert space operator with normal-like
features, are studied. Such operators include normal operators, k-normal
operators, subnormal operators, and Toeplitz operators. Our main result is the
determination of the noncommutative Choquet boundary for an operator system
generated by an irreducible periodic weighted unilateral shift operator.
