An n -tuple (S 1 , ... , S n − 1 , P) of commuting Hilbert space operators, for which the closed symmetrized polydisc Γ n = { (∑ i = 1 n z i , ∑ 1 ≤ i < j ≤ n z i z j , ... , ∏ i = 1 n z i) : | z i | ≤ 1 , i = 1 , ... , n } is a spectral set, is called a Γ n -contraction. Costara characterized a point in Γ n (or in its interior G n) via a rational function f ˜ s _ in the following way: s _ = (s 1 , ... , s n − 1 , p) ∈ Γ n (or ∈ G n) ⇔ | f ˜ s _ (z) | ≤ 1 (or < 1) , ∀ z ∈ D ‾ , where f ˜ s _ is the continuous extension of Costara's function f s _ defined on D by f s _ (z) = n (− 1) n p z n − 1 + (n − 1) (− 1) n − 1 s n − 1 z n − 2 + ... + (− s 1) n − (n − 1) s 1 z + ... + (− 1) n − 1 s n − 1 z n − 1 . In this paper, we investigate an operator theoretic analogue of this result. We prove that for a Γ n -contraction only the forward implication holds and the converse holds for subnormal Γ n -contractions. We provide a counterexample to show that the converse does not hold if we omit the subnormality condition. We obtain a few new characterizations of Γ n -isometries and Γ n -unitaries, which are analogues for Γ n -contractions of the isometries and unitaries respectively in one variable operator theory. We show that (s 1 , ... , s n − 1 , p) ∈ Γ n if and only if (s 1 , ... , s n − 1 , p , 0 , ... , 0 ︸ k − t i m e s) ∈ Γ n + k , but, the operator theoretic analogue of this result does not hold in general. We prove that if (S 1 , ... , S n − 1 , P) is a Γ n -contraction then (S 1 , ... , S n − 1 , P , 0 , ... , 0 ︸ k − t i m e s) is a Γ n + k -contraction and construct a counterexample to show that the converse is not always true. [ABSTRACT FROM AUTHOR]