A Note on Tetrablock Contractions

A commuting triple of operators $(A,B,P)$ on a Hilbert space $\mathcal{H}$ is called a tetrablock contraction if the closure of the set $$E = \{\underline{x}=(x_1,x_2,x_3)\in \mathbb{C}^3: 1-x_1z-x_2w+x_3zw \neq 0 \text{whenever}|z| \leq 1\text{and}|w| \leq 1 \}$$ is a spectral set. In this paper, we have constructed a functional model and produced a complete unitary invariant for a pure tetrablock contraction. In this construction, the fundamental operators, which are the unique solutions of the operator equations $$A-B^*P = D_PX_1D_P \text{and} B-A^*P=D_PX_2D_P, \text{where $X_1,X_2 \in \mathcal{B}(\mathcal{D}_P)$},$$ play a big role. As a corollary to the functional model, we show that every pure tetrablock isometry $(A,B,P)$ on a Hilbert space $\mathcal{H}$ is unitarily equivalent to $(M_{G_1^*+G_2z}, M_{G_2^*+G_1z},M_z)$ on $H^2_{\mathcal{D}_{P^*}}(\mathbb{D})$, where $G_1$ and $G_2$ are the fundamental operators of $(A^*,B^*,P^*)$. We prove a Beurling-Lax-Halmos type theorem for a triple of operators $(M_{F_1^*+F_2z},M_{F_2^*+F_1z},M_z)$, where $\mathcal{E}$ is a Hilbert space and $F_1,F_2 \in \mathcal{B}(\mathcal{E})$. We deal with a natural example of tetrablock contraction on functions space to find out its fundamental operators.