Functions of pairs of unbounded noncommuting self-adjoint operators under perturbation

2022/07/06 by А. Б. Александров, Aleksandrov, Aleksei, Vladimir Peller +1
Mathematics · #Spectral Theory in Mathematical Physics #advanced mathematical theories #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2207.02983

Abstract

For a pair (A,B) of not necessarily bounded and not necessarily commuting self-adjoint operators and for a function f on the Euclidean space \Bbb R² that belongs to the inhomogeneous Besov class B_∞,1¹(\Bbb R²), we define the function f(A,B) of these operators as a densely defined operator. We consider the problem of estimating the functions f(A,B) under perturbations of the pair (A,B). It is established that if 1≤ p≤2, and (A₁,B₁) and (A₂,B₂) are pairs of not necessarily bounded and not necessarily commuting self-adjoint operators such that the operators A₁-A₂ and B₁-B₂ belong to the Schatten--von Neumann class \boldsymbolS_p with p∈[1,2] and f∈ B_∞,1¹(\Bbb R²), then the following Lipschitz type estimate holds: ‖f(A₁,B₁)-f(A₂,B₂)‖_\boldsymbolS_p \leconst‖f‖_B_∞,1¹max\‖A₁-A₂‖_\boldsymbolS_p,‖B₁-B₂‖_\boldsymbolS_p\.

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