A commuting triple of operators (A,B,P) on a Hilbert space H is called a tetrablock contraction if the closure of the set E={(a₁₁,a₂₂,detA): A=\begin{pmatrix} a₁₁ & a₁₂ a₂₁ & a₂₂ \end{pmatrix} with ||A|| <1} is a spectral set. In this paper, we construct a functional model and produce a set of complete unitary invariants for a pure tetrablock contraction. In this construction, the fundamental operators, which are the unique solutions of the operator equations A-B*P = D_PX₁D_P and B-A*P=D_PX₂D_P, where X₁,X₂ ∈ B(D_P) play a pivotal role. As a result of the functional model, we show that every pure tetrablock isometry (A,B,P) on an abstract Hilbert space H is unitarily equivalent to the tetrablock contraction (M_G1*+G2z, M_G2*+G1z,M_z) on H²_DP*(D), where G₁ and G₂ are the fundamental operators of (A*,B*,P*). We prove a Beurling-Lax-Halmos type theorem for a triple of operators (M_F1*+F2z,M_F2*+F1z,M_z), where E is a Hilbert space and F₁,F₂ ∈ B(E). We also deal with a natural example of tetrablock contraction on a functions space to find out its fundamental operators.

The author's research is supported by University Grants Commission Center for Advanced Studies.