A characterization of compactness via bilinear $T1$ theorem

We establish a bilinear $T1$ theorem to characterize the weighted compactness of bilinear Calder\'{o}n--Zygmund operators. Let T be a bilinear operator associated with a standard bilinear Calder\'{o}n--Zygmund kernel. We demonstrate that T can be extended to a compact bilinear operator from Lp₁(w₁p₁) × Lp₂(w₂p₂) to Lᵖ(wᵖ) for all exponents 1/p = 1/p₁ + 1/p₂ with 1<p₁, p₂< ∞ and for all weights (w₁, w₂) ∈ A(p₁, p₂) if and only if the following conditions hold: (i) T is associated with a compact bilinear Calder\'{o}n--Zygmund kernel, (ii) T satisfies the weak compactness property, and (iii) T(1,1), T*1(1,1), T*2(1,1) ∈ CMO(Rⁿ).