Weak type operator Lipschitz and commutator estimates for commuting tuples

2017/03/09 by Caspers, Martijn, Sukochev, Fedor, Zanin, Dmitriy · 2 citations
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1703.03089

Abstract

Let f: ℝ^d →ℝ be a Lipschitz function. If B is a bounded self-adjoint operator and if \A_k\_k=1^d are commuting bounded self-adjoint operators such that [A_k,B]∈ L₁(H), then ‖[f(A₁,⋯,A_d),B]‖_1,∞≤ c(d)‖∇(f)‖_∞max_{1≤ k≤ d}‖[A_k,B]‖₁, where c(d) is a constant independent of f, M and A,B and ‖⋅‖_1,∞ denotes the weak L₁-norm. If \X_k\_k=1^d (respectively, \Y_k\_k=1^d) are commuting bounded self-adjoint operators such that X_k-Y_k∈ L₁(H), then ‖f(X₁,⋯,X_d)-f(Y₁,⋯,Y_d)‖_1,∞≤ c(d)‖∇(f)‖_∞max_{1≤ k≤ d}‖X_k-Y_k‖₁.

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