Functions of perturbed pairs of noncommutative dissipative operators

Abstract

Let a function f f belong to the inhomogeneous analytic Besov space ( B ∞ , 1 1 ) + ( R 2 ) (B_{\infty ,1}^{\,1})_+(\mathbb {R}^2) . For a pair ( L , M ) (L,M) of not necessarily commuting maximal dissipative operators, the function f ( L , M ) f(L,M) of L L and M M is defined as a densely defined linear operator. For p ∈ [ 1 , 2 ] p\in [1,2] , it is proved that if ( L 1 , M 1 ) (L_1,M_1) and ( L 2 , M 2 ) (L_2,M_2) are pairs of not necessarily commuting maximal dissipative operators such that both differences L 1 − L 2 L_1-L_2 and M 1 − M 2 M_1-M_2 belong to the Schatten–von Neumann class S p {\boldsymbol S}_p , then for an arbitrary function f f in (\mathcyr {B}_{\infty ,1}^{\,1})_+(\mathbb {R}^2), the operator difference f ( L 1 , M 1 ) − f ( L 2 , M 2 ) f(L_1,M_1)-f(L_2,M_2) belongs to S p {\boldsymbol S}_p and the following Lipschitz type estimate holds: \begin{equation*} \|f(L_1,M_1)-f(L_2,M_2)\|_{{\boldsymbol S}_p} \le const\|f\|_{\mathcyr {B}_{\infty ,1}^{\,1}}\max \big \{\|L_1-L_2\|_{{\boldsymbol S}_p},\|M_1-M_2\|_{{\boldsymbol S}_p}\big \}. \end{equation*}