Lipschitz functions,Schatten ideals,and unbounded derivations

It is proved that, for any Lipschitz function f(t ₁, ..., t _n ) of n variables, the corresponding map f _op: (A ₁, ...,A _n ) → f(A ₁, ..., A _n ) on the set of all commutative n-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal S ^p , p ∈ (1,∞). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz functions of one variable preserve domains of closed derivations with values in S ^p . It is also proved that the map f _op is Fréchet differentiable in the norm of S ^p if f is continuously differentiable.