Functons of perturbed pairs of dissipative operators

Abstract

Let $f$ be a function in the inhomogeneous analytic Besov space $B_{\infty ,1}^1$. For a pair $(L,M)$ of not necessarily commuting maximal dissipative operators, we define the function $f(L,M)$ of $L$ and $M$ as a densely defined linear operator. We prove for $p\in [1,2]$ that if $(L_1,M_1)$ and $(L_2,M_2)$ are pairs of not necessarily commuting maximal dissipative operators such that both differences $L_1-L_2$ and $M_1-M_2$ belong to the Schatten--von Neumann class ${\mathbf{S}}_p$ than for an arbitrary function $f$ in the inhomogeneous analytic Besov space $B_{\infty ,1}^1$, the operator difference $f(L_1,M_1)-f(L_2,M_2)$ belongs to ${\mathbf{S}}_p$ and the following Lipschitz type estimate holds: $$\Vert f(L_1,M_1)-f(L_2,M_2){\Vert }_{{\mathbf{S}}_p}\le \mathrm{const}\Vert f{\Vert }_{B_{\infty ,1}^1}\max \{\Vert L_1-L_2{\Vert }_{{\mathbf{S}}_p},\Vert M_1-M_2{\Vert }_{{\mathbf{S}}_p}\}.$$

arXiv comment18 pages. arXiv admin note: substantial text overlap with arXiv:2109.02339