For a commuting pair T of bounded linear operators T₁ and T₂ on a Hilbert space H, let D_T = T₂*T₂ -T₁*T₁. If T₂* D_T T₂ ≤ D_T and the Taylor spectrum of T is contained in the Hartogs triangle Δ_H, then for any bounded holomorphic function φ on Δ_H, ||φ(T)|| ≤ ||φ||_∞. We deduce this fact from an analogue of von Neumann's inequality for bounded domains in C^d. The proof of the latter closely follows the model theory approach as developed in [1].

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