Operator Lipschitz estimates in the unitary setting

We develop a Lipschitz estimate for unitary operators. More specifically, we show that for each p ∈ ( 1 , ∞ ) p\in (1,\infty ) there exists a constant d p d_p such that ‖ f ( U ) − f ( V ) ‖ p ≤ d p ‖ U − V ‖ p \left \Vert f(U) - f(V)\right \Vert _p \leq d_p \left \Vert U - V\right \Vert _p for all Lipschitz functions f : T → C f: \mathbb {T} \to \mathbb {C} and unitary operators U U and V V .